[
{
"id": "authors:ynbpx-c4633",
"collection": "authors",
"collection_id": "ynbpx-c4633",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230725-500420000.2",
"type": "article",
"title": "Sums of transcendental dilates",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Lim",
"given_name": "Jeck",
"clpid": "Lim-Jeck"
}
],
"abstract": "We show that there is an absolute constant c > 0 such that |A + \u03bb \u2022 A| \u2a7e e^(c\u221alog|A)| |A| for any finite subset A of \u211d and any transcendental number \u03bb \u2208 \u211d. By a construction of Konyagin and \u0141aba, this is best possible up to the constant c.",
"doi": "10.1112/blms.12870",
"issn": "0024-6093",
"publisher": "London Mathematical Society",
"publication": "Bulletin of the London Mathematical Society",
"publication_date": "2023-08-18"
},
{
"id": "authors:z589a-6dp27",
"collection": "authors",
"collection_id": "z589a-6dp27",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230717-55915200.33",
"type": "article",
"title": "On the size-Ramsey number of grids",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Nenadov",
"given_name": "Rajko",
"clpid": "Nenadov-Rajko"
},
{
"family_name": "Truji\u0107",
"given_name": "Milo\u0161",
"orcid": "0000-0002-7592-3630",
"clpid": "Truji\u0107-Milo\u0161"
}
],
"abstract": "We show that the size-Ramsey number of the \u221an \u00d7 \u221an grid graph is O(n^(5/4)), improving a previous bound of n^(3/2 + o(1)) by Clemens, Miralaei, Reding, Schacht, and Taraz.",
"doi": "10.1017/s0963548323000147",
"issn": "0963-5483",
"publisher": "Cambridge University Press",
"publication": "Combinatorics, Probability and Computing",
"publication_date": "2023-08-16"
},
{
"id": "authors:hjwhg-sd610",
"collection": "authors",
"collection_id": "hjwhg-sd610",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221031-575177800.7",
"type": "article",
"title": "A new bound for the Brown-Erd\u0151s-S\u00f3s problem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Gishboliner",
"given_name": "Lior",
"orcid": "0000-0003-0688-8111",
"clpid": "Gishboliner-Lior"
},
{
"family_name": "Levanzov",
"given_name": "Yevgeny",
"clpid": "Levanzov-Yevgeny"
},
{
"family_name": "Shapira",
"given_name": "Asaf",
"clpid": "Shapira-Asaf"
}
],
"abstract": "Let f(n, v, e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e \u2265 3, what is the smallest integer d = d(e) such that f(n, e+d, e) = o(n\u00b2)? This question has its origins in work of Brown, Erd\u0151s and S\u00f3s from the early 70's and the standard conjecture is that d(e) = 3 for every e \u2265 3. The state of the art result regarding this problem was obtained in 2004 by S\u00e1rk\u00f6zy and Selkow, who showed that f(n, e+2+[log\u2082e], e) = o(n\u00b2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the S\u00e1rk\u00f6zy\u2013Selkow bound, showing that f(n, e+O(log e / log log e), e) = o(n\u00b2).",
"doi": "10.1016/j.jctb.2022.08.005",
"issn": "0095-8956",
"publisher": "Elsevier",
"publication": "Journal of Combinatorial Theory. Series B",
"publication_date": "2023-01",
"series_number": "2",
"volume": "158",
"issue": "2",
"pages": "1-35"
},
{
"id": "authors:v1t30-85j35",
"collection": "authors",
"collection_id": "v1t30-85j35",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221011-128968500.8",
"type": "article",
"title": "Threshold Ramsey multiplicity for paths and even cycles",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-Jacob"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-Benny"
},
{
"family_name": "Wei",
"given_name": "Fan",
"clpid": "Wei-Fan"
}
],
"abstract": "The Ramsey number r(H) of a graph H is the minimum integer such that any two-coloring of the edges of the complete graph K\u2099 contains a monochromatic copy of H. While this definition only asks for a single monochromatic copy of H, it is often the case that every two-edge-coloring of the complete graph on r(H) vertices contains many monochromatic copies of H. The minimum number of such copies over all two-colorings of K_(r(H)) will be referred to as the threshold Ramsey multiplicity of H. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant c such that the threshold Ramsey multiplicity of a path or an even cycle on k vertices is at least (ck)\u1d4f. This bound is tight up to the constant c. We prove a similar result for odd cycles in a companion paper.",
"doi": "10.1016/j.ejc.2022.103612",
"issn": "0195-6698",
"publisher": "Elsevier",
"publication": "European Journal of Combinatorics",
"publication_date": "2023-01",
"volume": "107",
"pages": "Art. No. 103612"
},
{
"id": "authors:3bzbc-xj302",
"collection": "authors",
"collection_id": "3bzbc-xj302",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221017-10817000.4",
"type": "article",
"title": "The upper logarithmic density of monochromatic subset sums",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-Jacob"
},
{
"family_name": "Pham",
"given_name": "Huy Tuan",
"clpid": "Pham-Huy-Tuan"
}
],
"abstract": "We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least (2 + \u221a3)/4 and this is best possible. This answers a 40-year-old question of Erd\u0151s.",
"doi": "10.1112/mtk.12167",
"issn": "0025-5793",
"publisher": "University College London",
"publication": "Mathematika",
"publication_date": "2022-10",
"series_number": "4",
"volume": "68",
"issue": "4",
"pages": "1292-1301"
},
{
"id": "authors:kd92j-5zw50",
"collection": "authors",
"collection_id": "kd92j-5zw50",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221107-997760900.3",
"type": "article",
"title": "Threshold Ramsey multiplicity for odd cycles",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-Jacob"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-Benny"
},
{
"family_name": "Wei",
"given_name": "Fan",
"clpid": "Wei-Fan"
}
],
"abstract": "The Ramsey number r(H) of a graph H is the minimum n such that any two-coloring of the edges of the complete graph K\u2099 contains a monochromatic copy of H. The threshold Ramsey multiplicity m(H) is then the minimum number of monochromatic copies of H taken over all two-edge-colorings of K_(r(H)). The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant c such that the threshold Ramsey multiplicity for a path or even cycle with k vertices is at least (ck)\u1d4f, which is tight up to the value of c. Here, using different methods, we show that the same result also holds for odd cycles with k vertices.",
"doi": "10.33044/revuma.2874",
"issn": "1669-9637",
"publisher": "Union Matematica Argentina",
"publication": "Revista de la Uni\u00f3n Matem\u00e1tica Argentina",
"publication_date": "2022-08",
"series_number": "1",
"volume": "64",
"issue": "1",
"pages": "49-68"
},
{
"id": "authors:63bb7-54c82",
"collection": "authors",
"collection_id": "63bb7-54c82",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220726-997455000",
"type": "article",
"title": "Ramsey numbers of trails and circuits",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Tyomkyn",
"given_name": "Mykhaylo",
"clpid": "Tyomkyn-Mykhaylo"
}
],
"abstract": "We show that every two-colouring of the edges of the complete graph K\u2099 contains a monochromatic trail or circuit of length at least 2n\u00b2/9+o(n\u00b2), which is asymptotically best possible.",
"doi": "10.1002/jgt.22865",
"issn": "0364-9024",
"publisher": "Wiley",
"publication": "Journal of Graph Theory",
"publication_date": "2022-07-27"
},
{
"id": "authors:tg42p-xc421",
"collection": "authors",
"collection_id": "tg42p-xc421",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220718-901273500",
"type": "article",
"title": "The size\u2010Ramsey number of cubic graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Nenadov",
"given_name": "Rajko",
"clpid": "Nenadov-Rajko"
},
{
"family_name": "Truji\u0107",
"given_name": "Milo\u0161",
"clpid": "Truji\u0107-Milo\u0161"
}
],
"abstract": "We show that the size-Ramsey number of any cubic graph with n vertices is O(n^(8/5)), improving a bound of n^(5/3+o(1)) due to Kohayakawa, R\u00f6dl, Schacht, and Szemer\u00e9di. The heart of the argument is to show that there is a constant C such that a random graph with Cn vertices where every edge is chosen independently with probability p\u2a7eC_n^(\u22122/5) is with high probability Ramsey for any cubic graph with n vertices. This latter result is best possible up to the constant.",
"doi": "10.1112/blms.12682",
"issn": "0024-6093",
"publisher": "London Mathematical Society",
"publication": "Bulletin of the London Mathematical Society",
"publication_date": "2022-07-20"
},
{
"id": "authors:cq695-mc662",
"collection": "authors",
"collection_id": "cq695-mc662",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200914-085046091",
"type": "article",
"title": "Some remarks on the Zarankiewicz problem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1's in an m \u00d7 n matrix with all entries 0 or 1 containing no s \u00d7 t submatrix consisting entirely of 1's. We show that a classical upper bound for z(m, n; s, t) due to K\u0151v\u00e1ri, S\u00f3s and Tur\u00e1n is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.",
"doi": "10.1017/S0305004121000475",
"issn": "0305-0041",
"publisher": "Cambridge University Press",
"publication": "Mathematical Proceedings of the Cambridge Philosophical Society",
"publication_date": "2022-07",
"series_number": "1",
"volume": "173",
"issue": "1",
"pages": "155-161"
},
{
"id": "authors:fstjd-qqb57",
"collection": "authors",
"collection_id": "fstjd-qqb57",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170936221",
"type": "article",
"title": "Monochromatic combinatorial lines of length three",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We show that there is a positive constant c such that any colouring of the cube [3]^n in c log log n colours contains a monochromatic combinatorial line.",
"doi": "10.1090/proc/15739",
"issn": "0002-9939",
"publisher": "American Mathematical Society",
"publication": "Proceedings of the American Mathematical Society",
"publication_date": "2022-01",
"series_number": "1",
"volume": "150",
"issue": "1",
"pages": "1-4"
},
{
"id": "authors:5bfv1-vtn47",
"collection": "authors",
"collection_id": "5bfv1-vtn47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230515-296699000.2",
"type": "article",
"title": "Which graphs can be counted in C\u2084-free graphs?",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"orcid": "0000-0002-0664-497X",
"clpid": "Fox-Jacob"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"orcid": "0000-0003-3307-9475",
"clpid": "Sudakov-Benny"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"orcid": "0000-0002-1995-3755",
"clpid": "Zhao-Yufei"
}
],
"abstract": "For which graphs F is there a sparse F-counting lemma in C\u2084-free graphs? We are interested in identifying graphs F with the property that, roughly speaking, if G is an n-vertex C\u2084-free graph with on the order of n^(3/2) edges, then the density of F in G, after a suitable normalization, is approximately at least the density of F in an \u03b5-regular approximation of G. In recent work, motivated by applications in extremal and additive combinatorics, we showed that C\u2085 has this property. Here we construct a family of graphs with the property.",
"doi": "10.4310/pamq.2022.v18.n6.a4",
"issn": "1558-8599",
"publisher": "International Press",
"publication": "Pure and Applied Mathematics Quarterly",
"publication_date": "2022",
"series_number": "6",
"volume": "18",
"issue": "6",
"pages": "2413-2432"
},
{
"id": "authors:137hj-tjx44",
"collection": "authors",
"collection_id": "137hj-tjx44",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200914-101307280",
"type": "article",
"title": "The regularity method for graphs with few 4-cycles",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-Jacob"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-Benny"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: \n\nEvery n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n^(3/2)) edges. \nFor r \u2a7e 3, every n-vertex r-graph with girth greater than 5 has o(n^(3/2)) edges. \nEvery subset of [n] without a nontrivial solution to the equation x\u2081 + x\u2082 + 2x\u2083 = x\u2084 + 3x\u2085 has size o(\u221an).",
"doi": "10.1112/jlms.12500",
"issn": "0024-6107",
"publisher": "Wiley",
"publication": "Journal of the London Mathematical Society",
"publication_date": "2021-12",
"series_number": "5",
"volume": "104",
"issue": "5",
"pages": "2376-2401"
},
{
"id": "authors:ypmvs-51734",
"collection": "authors",
"collection_id": "ypmvs-51734",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200914-134941914",
"type": "article",
"title": "Repeated Patterns in Proper Colorings",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Tyomkyn",
"given_name": "Mykhaylo",
"clpid": "Tyomkyn-Mykhaylo"
}
],
"abstract": "For a fixed graph H, what is the smallest number of colors C such that there is a proper edge-coloring of the complete graph K_n with C colors containing no two vertex-disjoint color-isomorphic copies, or repeats, of H? We study this function and its generalization to more than two copies using a variety of combinatorial, probabilistic, and algebraic techniques. For example, we show that for any tree T there exists a constant c such that any proper edge-coloring of K_n with at most c n^2 colors contains two repeats of T, while there are colorings with at most c' n^(3/2) colors for some absolute constant c' containing no three repeats of any tree with at least two edges. We also show that for any graph H containing a cycle there exist k and c such that there is a proper edge-coloring of K_n with at most c n colors containing no k repeats of H, while for a tree T with m edges, a coloring with o(n^((m+1)/m)) colors contains \u03c9(1) repeats of T.",
"doi": "10.1137/21M1414103",
"issn": "0895-4801",
"publisher": "Society for Industrial and Applied Mathematics",
"publication": "SIAM Journal on Discrete Mathematics",
"publication_date": "2021-09-28",
"series_number": "3",
"volume": "35",
"issue": "3",
"pages": "2249-2264"
},
{
"id": "authors:mzy87-2zv57",
"collection": "authors",
"collection_id": "mzy87-2zv57",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20211105-180659290",
"type": "article",
"title": "Random Multilinear Maps and the Erd\u0151s Box Problem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Pohoata",
"given_name": "Cosmin",
"orcid": "0000-0002-3757-2526",
"clpid": "Pohoata-Cosmin"
},
{
"family_name": "Zakharov",
"given_name": "Dmitriy",
"clpid": "Zakharov-Dmitriy"
}
],
"abstract": "By using random multilinear maps, we provide new lower bounds for the Erd\u0151s\nbox problem, the problem of estimating the extremal number of the complete d-partite duniform\nhypergraph with two vertices in each part, thereby improving on work of Gunderson,\nR\u00f6dl and Sidorenko.",
"doi": "10.19086/da.28336",
"issn": "2397-3129",
"publisher": "Scholastica",
"publication": "Discrete Analysis",
"publication_date": "2021-09-27",
"volume": "2021",
"pages": "Art. No. 17"
},
{
"id": "authors:sqbep-vc039",
"collection": "authors",
"collection_id": "sqbep-vc039",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170943053",
"type": "article",
"title": "More on the extremal number of subdivisions",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Janzer",
"given_name": "Oliver",
"clpid": "Janzer-Oliver"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
}
],
"abstract": "Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing K\u2032_(s,t) for the subdivision of the bipartite graph K_(s,t), we show that ex(n,K\u2032_(s,t)) = O(n^(3/2)\u22121/2s)). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s,k \u2265 1, we show that ex(n,L) = \u0398(n^(1+s/sk+1)) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erd\u0151s and Simonovits, which asserts that every rational number r \u2208 (1, 2) is realisable in the sense that ex(n, H) = \u0398(n^r) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k \u2265 1. Writing H^k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists \u03b4 > 0 such that ex(n, H^(k\u22121)) = O(^(n1+1/k\u2212\u03b4)), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C\u2084 as a subgraph satisfies ex(n, H) = o(n^(2\u22121/r)).",
"doi": "10.1007/s00493-020-4202-1",
"issn": "0209-9683",
"publisher": "Springer",
"publication": "Combinatorica",
"publication_date": "2021-08",
"series_number": "4",
"volume": "41",
"issue": "4",
"pages": "465-494"
},
{
"id": "authors:bepy3-mey72",
"collection": "authors",
"collection_id": "bepy3-mey72",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230606-676918000.2",
"type": "monograph",
"title": "Which graphs can be counted in C\u2084-free graphs?",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"orcid": "0000-0002-0664-497X",
"clpid": "Fox-Jacob"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"orcid": "0000-0003-3307-9475",
"clpid": "Sudakov-Benny"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"orcid": "0000-0002-1995-3755",
"clpid": "Zhao-Yufei"
}
],
"abstract": "For which graphs F is there a sparse F-counting lemma in C\u2084-free graphs? We are interested in identifying graphs F with the property that, roughly speaking, if G is an n-vertex C\u2084-free graph with on the order of n^(3/2) edges, then the density of F in G, after a suitable normalization, is approximately at least the density of F in an \u03b5-regular approximation of G. In recent work, motivated by applications in extremal and additive combinatorics, we showed that C\u2085 has this property. Here we construct a family of graphs with the property.",
"publisher": "arXiv",
"publication_date": "2021-06-06"
},
{
"id": "authors:8w53p-t0j17",
"collection": "authors",
"collection_id": "8w53p-t0j17",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210602-132638631",
"type": "article",
"title": "Extremal Numbers of Cycles Revisited",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We give a simple geometric interpretation of an algebraic construction of Wenger that gives n-vertex graphs with no cycle of length 4, 6, or 10 and close to the maximum number of edges.",
"doi": "10.1080/00029890.2021.1886845",
"issn": "0002-9890",
"publisher": "Mathematical Association of America",
"publication": "American Mathematical Monthly",
"publication_date": "2021-05-28",
"series_number": "5",
"volume": "128",
"issue": "5",
"pages": "464-466"
},
{
"id": "authors:5k8hv-4ts49",
"collection": "authors",
"collection_id": "5k8hv-4ts49",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201216-111908862",
"type": "article",
"title": "Lower bounds for multicolor Ramsey numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Ferber",
"given_name": "Asaf",
"clpid": "Ferber-Asaf"
}
],
"abstract": "We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.",
"doi": "10.1016/j.aim.2020.107528",
"issn": "0001-8708",
"publisher": "Elsevier",
"publication": "Advances in Mathematics",
"publication_date": "2021-02-12",
"volume": "378",
"pages": "Art. No. 107528"
},
{
"id": "authors:f781n-d0219",
"collection": "authors",
"collection_id": "f781n-d0219",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190820-162204728",
"type": "article",
"title": "Ramsey games near the critical threshold",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Das",
"given_name": "Shagnik",
"clpid": "Das-Shagnik"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
},
{
"family_name": "M\u00e9sz\u00e1ros",
"given_name": "Tam\u00e1s",
"clpid": "M\u00e9sz\u00e1ros-T"
}
],
"abstract": "A well\u2010known result of R\u00f6dl and Ruci\u0144ski states that for any graph H there exists a constant C such that if p \u2265 Cn^(-1/m2(H)), then the random graph G_(n,\u2009p) is a.a.s. H\u2010Ramsey, that is, any 2\u2010coloring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0\u2010statement also holds, that is, there exists c\u2009>\u20090 such that whenever p \u2264 Cn^(-1/m2(H)) the random graph Gn,\u2009p is a.a.s. not H\u2010Ramsey. We show that near this threshold, even when G_(n,\u2009p) is not H\u2010Ramsey, it is often extremely close to being H\u2010Ramsey. More precisely, we prove that for any constant c\u2009>\u20090 and any strictly 2\u2010balanced graph H, if p \u2265 Cn^(-1/m2(H)), then the random graph G_(n,\u2009p) a.a.s. has the property that every 2\u2010edge\u2010coloring without monochromatic copies of H cannot be extended to an H\u2010free coloring after \u03c9(1) extra random edges are added. This generalizes a result by Friedgut, Kohayakawa, R\u00f6dl, Ruci\u0144ski, and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three\u2010color case and show that these theorems need not hold when H is not strictly 2\u2010balanced.",
"doi": "10.1002/rsa.20959",
"issn": "1042-9832",
"publisher": "Wiley",
"publication": "Random Structures and Algorithms",
"publication_date": "2020-12",
"series_number": "4",
"volume": "57",
"issue": "4",
"pages": "940-957"
},
{
"id": "authors:scafr-q3881",
"collection": "authors",
"collection_id": "scafr-q3881",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200909-070916091",
"type": "article",
"title": "Short proofs of some extremal results\u00a0III",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.",
"doi": "10.1002/rsa.20953",
"issn": "1042-9832",
"publisher": "Wiley",
"publication": "Random Structures & Algorithms",
"publication_date": "2020-12",
"series_number": "4",
"volume": "57",
"issue": "4",
"pages": "958-982"
},
{
"id": "authors:hbkkk-d8404",
"collection": "authors",
"collection_id": "hbkkk-d8404",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170932806",
"type": "article",
"title": "Hypergraph expanders of all uniformities from Cayley graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Tidor",
"given_name": "Jonathan",
"clpid": "Tidor-J"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "Hypergraph expanders are hypergraphs with surprising, non\u2010intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of 3\u2010uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of r\u2010uniform hypergraph expanders for all r \u2a7e 3.",
"doi": "10.1112/plms.12371",
"issn": "0024-6115",
"publisher": "London Mathematical Society",
"publication": "Proceedings of the London Mathematical Society",
"publication_date": "2020-11",
"series_number": "5",
"volume": "121",
"issue": "5",
"pages": "1311-1336"
},
{
"id": "authors:eyaza-xse95",
"collection": "authors",
"collection_id": "eyaza-xse95",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200915-144809437",
"type": "monograph",
"title": "A New Bound for the Brown-Erd\u0151s-S\u00f3s Problem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Gishboliner",
"given_name": "Lior",
"orcid": "0000-0003-0688-8111",
"clpid": "Gishboliner-Lior"
},
{
"family_name": "Levanzov",
"given_name": "Yevgeny",
"clpid": "Levanzov-Yevgeny"
},
{
"family_name": "Shapira",
"given_name": "Asaf",
"clpid": "Shapira-Asaf"
}
],
"abstract": "Let f(n,v,e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e\u22653, what is the smallest integer d=d(e) so that f(n,e+d,e)=o(n\u00b2)? This question has its origins in work of Brown, Erd\u0151s and S\u00f3s from the early 70's and the standard conjecture is that d(e)=3 for every e\u22653. The state of the art result regarding this problem was obtained in 2004 by S\u00e1rk\u00f6zy and Selkow, who showed that f(n,e+2+\u230alog\u2082e\u230b,e)=o(n\u00b2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the S\u00e1rk\u00f6zy--Selkow bound, showing that\nf(n,e+O(loge/logloge),e)=o(n\u00b2).",
"doi": "10.48550/arXiv.1912.08834",
"publisher": "arXiv",
"publication_date": "2020-09-15"
},
{
"id": "authors:pqpqt-9ba35",
"collection": "authors",
"collection_id": "pqpqt-9ba35",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200914-140122041",
"type": "article",
"title": "Ramsey Numbers of Books and Quasirandomness",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-Jacob"
},
{
"family_name": "Wigderson",
"given_name": "Yuval",
"orcid": "0000-0001-5909-9250",
"clpid": "Wigderson-Yuval"
}
],
"abstract": "The book graph B^((k))_n consists of n copies of K_(k+1) joined along a common K_k. The Ramsey numbers of B^((k))_n are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed k, thus answering an old question of Erd\u0151s, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemer\u00e9di's regularity lemma, thus providing much tighter control on the error term. Finally, we prove a conjecture of Nikiforov, Rousseau, and Schelp by showing that all extremal colorings for this Ramsey problem are quasirandom.",
"doi": "10.1007/s00493-021-4409-9",
"issn": "0209-9683",
"publisher": "Springer",
"publication": "Combinatorica",
"publication_date": "2020-09-14"
},
{
"id": "authors:n7nss-psj53",
"collection": "authors",
"collection_id": "n7nss-psj53",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170946478",
"type": "article",
"title": "Books versus triangles at the extremal density",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "A celebrated result of Mantel shows that every graph on n vertices with [n\u00b2/4] + 1 edges must contain a triangle. A robust version of this result, due to Rademacher, says that there\nmust, in fact, be at least [n/2] triangles in any such graph. Another strengthening, due to the\ncombined efforts of many authors starting with Erd\u0151s, says that any such graph must have an edge\nwhich is contained in at least n/6 triangles. Following Mubayi, we study the interplay between\nthese two results, that is, between the number of triangles in such graphs and their book number,\nthe largest number of triangles sharing an edge. Among other results, Mubayi showed that for any\n1/6 \u2264 \u03b2 < 1/4 there is \u03b3 > 0 such that any graph on n vertices with at least [n\u00b2/4] +1 edges and book number at most \u03b2n contains at least (\u03b3 - o(1))n\u00b3 triangles. He also asked for a more precise\nestimate for \u03b3 in terms of \u03b2. We make a conjecture about this dependency and prove this conjecture\nfor \u03b2 = 1/6 and for 0.2495 \u2264 \u03b2 < 1/4, thereby answering Mubayi's question in these ranges.",
"doi": "10.1137/19M1261766",
"issn": "0895-4801",
"publisher": "Society for Industrial and Applied Mathematics",
"publication": "SIAM Journal on Discrete Mathematics",
"publication_date": "2020-02-12",
"series_number": "1",
"volume": "34",
"issue": "1",
"pages": "385-398"
},
{
"id": "authors:7e9nc-jny65",
"collection": "authors",
"collection_id": "7e9nc-jny65",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190820-161459631",
"type": "monograph",
"title": "Sidorenko's conjecture for higher tree decompositions",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Kim",
"given_name": "Jeong Han",
"clpid": "Kim-Jeong-Han"
},
{
"family_name": "Lee",
"given_name": "Choongbum",
"clpid": "Lee-Choongbum"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
}
],
"abstract": "This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.",
"publisher": "Caltech Library",
"publication_date": "2019-08-21"
},
{
"id": "authors:xam4m-fzc19",
"collection": "authors",
"collection_id": "xam4m-fzc19",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170939635",
"type": "monograph",
"title": "Independent arithmetic progressions",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We show that there is a positive constant c such that any graph on vertex set [n] with at most cn^(2)/k^(2) log k edges contains an independent set of order k whose vertices form an arithmetic progression. We also present applications of this result to several questions in Ramsey theory.",
"doi": "10.48550/arXiv.1901.05084",
"publication_date": "2019-08-20"
},
{
"id": "authors:427jk-z8f82",
"collection": "authors",
"collection_id": "427jk-z8f82",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170914451",
"type": "monograph",
"title": "Hypergraph cuts above the average",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Kwan",
"given_name": "Matthew",
"clpid": "Kwan-Matthew"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2 + \u03a9(\u221am), and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is \u03a9(\u221am) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k \u2265 4 or r \u2265 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is \u03a9(m^(5/9)) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.",
"doi": "10.48550/arXiv.1803.08462",
"publication_date": "2019-08-20"
},
{
"id": "authors:ew8zj-5j465",
"collection": "authors",
"collection_id": "ew8zj-5j465",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170928301",
"type": "monograph",
"title": "Sidorenko's conjecture for blow-ups",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
}
],
"abstract": "A celebrated conjecture of Sidorenko and Erd\u0151s-Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. \n\nOur contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A \u222a B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A \u222a B, there is a positive integer p such that the blow-up H_(A)^(p) formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. \n\nAnother way of viewing this latter result is that for every bipartite H there is a positive integer p such that an L^(p)-version of Sidorenko's conjecture holds for H.",
"doi": "10.48550/arXiv.1809.01259",
"publication_date": "2019-08-20"
},
{
"id": "authors:86jzm-jh215",
"collection": "authors",
"collection_id": "86jzm-jh215",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170911002",
"type": "monograph",
"title": "Intervals in the Hales-Jewett theorem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Kam\u010dev",
"given_name": "Nina",
"clpid": "Kam\u010dev-N"
}
],
"abstract": "The Hales-Jewett theorem states that for any m and r there exists an n such that any r-colouring of the elements of [m]^n contains a monochromatic combinatorial line. We study the structure of the wildcard set S \u2286 [n] which determines this monochromatic line, showing that when r is odd there are r-colourings of [3]^n where the wildcard set of a monochromatic line cannot be the union of fewer than r intervals. This is tight, as for n sufficiently large there are always monochromatic lines whose wildcard set is the union of at most r intervals.",
"doi": "10.48550/arXiv.1801.08919",
"publication_date": "2019-08-20"
},
{
"id": "authors:3fpfb-rxa32",
"collection": "authors",
"collection_id": "3fpfb-rxa32",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170907486",
"type": "monograph",
"title": "Hypergraph expanders from Cayley graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We present a simple mechanism, which can be randomised, for constructing sparse 3-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over \u2124^(t)_(2) and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.",
"doi": "10.48550/arXiv.1709.10006",
"publication_date": "2019-08-20"
},
{
"id": "authors:3j2gs-bf147",
"collection": "authors",
"collection_id": "3j2gs-bf147",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170853333",
"type": "monograph",
"title": "Graphs with few paths of prescribed length between any two vertices",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We use a variant of Bukh's random algebraic method to show that for every natural number k \u2265 2 there exists a natural number \u2113 such that, for every n, there is a graph with n vertices and \u03a9_(k)(n^(1 + 1/k)) edges with at most \u2113 paths of length k between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.",
"doi": "10.48550/arXiv.1411.0856",
"publication_date": "2019-08-20"
},
{
"id": "authors:cs1hz-ws508",
"collection": "authors",
"collection_id": "cs1hz-ws508",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170842963",
"type": "monograph",
"title": "Sidorenko's conjecture for a class of graphs: an exposition",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "A famous conjecture of Sidorenko and Erd\u0151s-Simonovits states that if H is a bipartite graph then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. The goal of this expository note is to give a short self-contained proof (suitable for teaching in class) of the conjecture if H has a vertex complete to all vertices in the other part.",
"doi": "10.48550/arXiv.1209.0184",
"publisher": "arXiv",
"publication_date": "2019-08-20"
},
{
"id": "authors:kzw07-h3e65",
"collection": "authors",
"collection_id": "kzw07-h3e65",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170921338",
"type": "monograph",
"title": "On the extremal number of subdivisions",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
}
],
"abstract": "One of the cornerstones of extremal graph theory is a result of F\u00fcredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and Cn^(2 - (1/r)) edges contains a copy of H. This result is tight up to the constant when H contains a copy of K_(r,s) with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which F\u00fcredi's result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C_(4)-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and \u03b4 such that every graph with n vertices and Cn^(3/2) - \u03b4) edges contains a copy of H. This answers a question of Erd\u0151s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.",
"doi": "10.48550/arXiv.1807.05008",
"publication_date": "2019-08-20"
},
{
"id": "authors:4jhes-wet92",
"collection": "authors",
"collection_id": "4jhes-wet92",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800",
"type": "monograph",
"title": "The Ramsey number of books",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We show that in every two-colouring of the edges of the complete graph K_N there is a monochromatic K_k which can be extended in at least (1 + o_(k)(1))2^(-k)N ways to a monochromatic K_(k+1). This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book B_n^(k) to be the graph consisting of n copies of K_(k+1) all sharing a common K_k, we show that the Ramsey number r(B_n^(k)) = 2^(k)n + o_(k)(n). In this form, our result answers a question of Erd\u0151s, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.",
"doi": "10.48550/arXiv.1808.03157",
"publication_date": "2019-08-20"
},
{
"id": "authors:zsmzm-xth94",
"collection": "authors",
"collection_id": "zsmzm-xth94",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170846378",
"type": "monograph",
"title": "Linear forms from the Gowers uniformity norm",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "This is a companion note to our paper 'A relative Szemer\u00e9di theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemer\u00e9di theorem for (r + 1)-term arithmetic progressions assuming a linear forms condition. Here we show how to replace this condition with an assumption about the Gowers uniformity norm U^r.",
"doi": "10.48550/arXiv.1305.5565",
"publisher": "arXiv",
"publication_date": "2019-08-20"
},
{
"id": "authors:s7d7k-cwd29",
"collection": "authors",
"collection_id": "s7d7k-cwd29",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170832267",
"type": "monograph",
"title": "A note on lower bounds for hypergraph Ramsey numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \n\nr_(3)(l, l, l) \u2265 2^(l^(c log log l)).\n\nThe old bound, due to Erd\u0151s and Hajnal, was \n\nr_(3)(l, l, l) \u2265 2^(cl^(2) log^(2) l).",
"doi": "10.48550/arXiv.0711.5004",
"publisher": "arXiv",
"publication_date": "2019-08-20"
},
{
"id": "authors:fn6wv-r1370",
"collection": "authors",
"collection_id": "fn6wv-r1370",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170900578",
"type": "monograph",
"title": "Hedgehogs are not colour blind",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "R\u00f6dl",
"given_name": "Vojt\u011bch",
"clpid": "R\u00f6dl-V"
}
],
"abstract": "We exhibit a family of 3-uniform hypergraphs with the property that their 2-colour Ramsey numbers grow polynomially in the number of vertices, while their 4-colour Ramsey numbers grow exponentially. This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.",
"doi": "10.48550/arXiv.1511.00563",
"publication_date": "2019-08-20"
},
{
"id": "authors:ptnxn-m7r48",
"collection": "authors",
"collection_id": "ptnxn-m7r48",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170849884",
"type": "monograph",
"title": "Large subgraphs without complete bipartite graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "In this note, we answer the following question of Foucaud, Krivelevich and Perarnau. What is the size of the largest K_(r,s)-free subgraph one can guarantee in every graph G with m edges? We also discuss the analogous problem for hypergraphs.",
"doi": "10.48550/arXiv.1401.6711",
"publisher": "arXiv",
"publication_date": "2019-08-20"
},
{
"id": "authors:p8dqv-h2k91",
"collection": "authors",
"collection_id": "p8dqv-h2k91",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170917920",
"type": "monograph",
"title": "Online Ramsey Numbers and the Subgraph Query Problem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Grinshpun",
"given_name": "Andrey",
"clpid": "Grinshpun-A"
},
{
"family_name": "He",
"given_name": "Xiaoyu",
"clpid": "He-Xiaoyu"
}
],
"abstract": "The (m,n)-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red K_m or a blue K_n using as few turns as possible. The online Ramsey number [equation; see abstract in PDF for details] is the minimum number of edges Builder needs to guarantee a win in the (m,n)-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \n\n[equation; see abstract in PDF for details]\n\nfor the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement \n\n[equation; see abstract in PDF for details]\n\nfor the off-diagonal case, where m \u2265 3 is fixed and n \u2192 \u221e. Using a different randomized Painter strategy, we prove that [equation; see abstract in PDF for details], determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for m \u2265 4.\n\nIn connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph H in a sufficiently large unknown Erd\u0151s-R\u00e9nyi random graph G(N,p) using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.",
"doi": "10.48550/arXiv.1806.09726",
"publication_date": "2019-08-20"
},
{
"id": "authors:ctm7x-9at88",
"collection": "authors",
"collection_id": "ctm7x-9at88",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000172",
"type": "article",
"title": "Tower-type bounds for unavoidable patterns in words",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "A word \u03c9 is said to contain the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of \u03c9. Bean, Ehrenfeucht, and McNulty and, independently, Zimin characterised the patterns P which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains P. Zimin's characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by Z_1 = x_1 and Z_n = Z_n \u2212 1x_(n)Z_(n) \u2212 1. We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function f(n, q), the least integer such that any word of length f(n, q) over an alphabet of size q contains Z_(n). When n = 3, the first nontrivial case, we determine f(n, q) up to a constant factor, showing that f(3, q) = \u0398(2_(q)q!).",
"doi": "10.1090/tran/7751",
"issn": "0002-9947",
"publisher": "American Mathematical Society",
"publication": "Transactions of the American Mathematical Society",
"publication_date": "2019-05-30"
},
{
"id": "authors:qqfbd-zpw07",
"collection": "authors",
"collection_id": "qqfbd-zpw07",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959449",
"type": "article",
"title": "Lines in Euclidean Ramsey Theory",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
}
],
"abstract": "Let \u2113_m be a sequence of m points on a line with consecutive points of distance one. For every natural number n, we prove the existence of a red/blue-coloring of E^n containing no red copy of \u2113_2 and no blue copy of \u2113_m for any m \u2265 2^(cn). This is best possible up to the constant c in the exponent. It also answers a question of Erd\u0151s et al. (J Comb Theory Ser A 14:341\u2013363, 1973). They asked if, for every natural number n, there is a set K \u2282 E^1 and a red/blue-coloring of E^n containing no red copy of \u2113_2 and no blue copy of K.",
"doi": "10.1007/s00454-018-9980-5",
"issn": "0179-5376",
"publisher": "Springer",
"publication": "Discrete and Computational Geometry",
"publication_date": "2019-01",
"series_number": "1",
"volume": "61",
"issue": "1",
"pages": "218-225"
},
{
"id": "authors:ysvq9-q6365",
"collection": "authors",
"collection_id": "ysvq9-q6365",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958339",
"type": "article",
"title": "Some advances on Sidorenko's conjecture",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Kim",
"given_name": "Jeong Han",
"clpid": "Kim-Jeong-Han"
},
{
"family_name": "Lee",
"given_name": "Choongbum",
"clpid": "Lee-Choongbum"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
}
],
"abstract": "A bipartite graph H is said to have Sidorenko's property if the probability that the uniform random mapping from V(H) to the vertex set of any graph G is a homomorphism is at least the product over all edges in H of the probability that the edge is mapped to an edge of G. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if H has Sidorenko's property, then the Cartesian product of H with an even cycle also has Sidorenko's property.",
"doi": "10.1112/jlms.12142",
"issn": "0024-6107",
"publisher": "London Mathematical Society",
"publication": "Journal of the London Mathematical Society",
"publication_date": "2018-12",
"series_number": "3",
"volume": "98",
"issue": "3",
"pages": "593-608"
},
{
"id": "authors:vpw4b-fp197",
"collection": "authors",
"collection_id": "vpw4b-fp197",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190821-110928639",
"type": "article",
"title": "Quasirandomness in Hypergraphs",
"author": [
{
"family_name": "Aigner-Horev",
"given_name": "Elad",
"clpid": "Aigner-Horev-E"
},
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "H\u00e0n",
"given_name": "Hi\u1ec7p",
"clpid": "H\u00e0n-Hi\u1ec7p"
},
{
"family_name": "Person",
"given_name": "Yury",
"clpid": "Person-Y"
},
{
"family_name": "Schacht",
"given_name": "Mathias",
"clpid": "Schacht-M"
}
],
"abstract": "A graph G is called quasirandom if it possesses typical properties of the corresponding random graph G(n, p) with the same edge density as G. A well-known theorem of Chung, Graham and Wilson states that, in fact, many such 'typical' properties are asymptotically equivalent and, thus, a graph G possessing one property immediately satisfies the others.\n\nIn recent years, more quasirandom graph properties have been found and extensions to hypergraphs have been explored. For the latter, however, there exist several distinct notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. The purpose of this paper is to give short purely combinatorial proofs of most of Towsner's results.",
"issn": "1077-8926",
"publisher": "Electronic Journal of Combinatorics",
"publication": "Electronic Journal of Combinatorics",
"publication_date": "2018-08-24",
"series_number": "3",
"volume": "25",
"issue": "3",
"pages": "Art. No. P3.34"
},
{
"id": "authors:xr8k2-9be11",
"collection": "authors",
"collection_id": "xr8k2-9be11",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959631",
"type": "article",
"title": "Hereditary quasirandomness without regularity",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "A result of Simonovits and S\u00f3s states that for any fixed graph H and any \u03f5 > 0 there exists \u03b4 > 0 such that if G is an n-vertex graph with the property that every S \u2286 V(G) contains p^(e(H))|S|^(v(H)) \u00b1 \u03b4n^(v(H)) labelled copies of H, then G is quasirandom in the sense that every S \u2286 V(G) contains \u00bdp|S|^2 \u00b1 \u03f5n^2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on \u03b4^(\u22121) which is a tower of twos of height polynomial in \u03f5^(\u22121). We give an alternative proof of this theorem which avoids the regularity lemma and shows that \u03b4 may be taken to be linear in \u03f5 when H is a clique and polynomial in \u03f5 for general H. This answers a problem raised by Simonovits and S\u00f3s.",
"doi": "10.1017/s0305004116001055",
"issn": "0305-0041",
"publisher": "Cambridge University Press",
"publication": "Mathematical Proceedings of the Cambridge Philosophical Society",
"publication_date": "2018-05",
"series_number": "3",
"volume": "164",
"issue": "3",
"pages": "385-399"
},
{
"id": "authors:fjteg-4ys50",
"collection": "authors",
"collection_id": "fjteg-4ys50",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000355",
"type": "article",
"title": "Rational exponents in extremal graph theory",
"author": [
{
"family_name": "Bukh",
"given_name": "Boris",
"clpid": "Bukh-B"
},
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "Given a family of graphs \u210c, the extremal number ex(n, \u210c) is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family \u210c as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs \u210c_r such that ex (n, \u210c_r) = \u0398(n^r). This solves a longstanding problem in the area of extremal graph theory.",
"doi": "10.4171/jems/798",
"issn": "1435-9855",
"publisher": "European Mathematical Society",
"publication": "Journal of the European Mathematical Society",
"publication_date": "2018",
"series_number": "7",
"volume": "20",
"issue": "7",
"pages": "1747-1757"
},
{
"id": "authors:ehfed-mgm13",
"collection": "authors",
"collection_id": "ehfed-mgm13",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000449",
"type": "article",
"title": "Quasirandomness in hypergraphs",
"author": [
{
"family_name": "Aigner-Horev",
"given_name": "Elad",
"clpid": "Aigner-Horev-E"
},
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "H\u00e0n",
"given_name": "Hi\u1ec7p",
"clpid": "H\u00e0n-Hi\u1ec7p"
},
{
"family_name": "Person",
"given_name": "Yury",
"clpid": "Person-Y"
},
{
"family_name": "Schacht",
"given_name": "Mathias",
"clpid": "Schacht-M"
}
],
"abstract": "A graph G is called quasirandom if it possesses typical properties of the corresponding random graph G(n, p) with the same edge density as G. A well-known theorem of Chung, Graham and Wilson states that, in fact, many such 'typical' properties are asymptotically equivalent and, thus, a graph G possessing one property immediately satisfies the others.\n\nIn recent years, more quasirandom graph properties have been found and extensions to hypergraphs have been explored. For the latter, however, there exist several distinct notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. The purpose of this paper is to give short purely combinatorial proofs of most of Towsner's results.",
"doi": "10.1016/j.endm.2017.06.015",
"issn": "1571-0653",
"publisher": "Elsevier",
"publication": "Electronic Notes in Discrete Mathematics",
"publication_date": "2017-08",
"volume": "61",
"pages": "13-19"
},
{
"id": "authors:85q64-q6758",
"collection": "authors",
"collection_id": "85q64-q6758",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957357",
"type": "article",
"title": "Finite reflection groups and graph norms",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Lee",
"given_name": "Joonkyung",
"clpid": "Lee-Joonkyung"
}
],
"abstract": "Given a graph H on vertex set {1, 2, \u2022 \u2022 \u2022, n} and a function f : [0, 1]^2 \u2192 \u211d, define [equation; see abstract in PDF for details], where \u03bc is the Lebesgue measure on [0, 1]. We say that H is norming if \u2225\u2022\u2225_H is a semi-norm. A similar notion \u2225\u2022\u2225_r(H) is defined by \u2225f\u2225_r(H) := \u2225|f|\u2225_H and H is said to be weakly norming if \u2225\u2022\u2225_r(H) is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorphisms is weakly norming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture.",
"doi": "10.1016/j.aim.2017.05.009",
"issn": "0001-8708",
"publisher": "Elsevier",
"publication": "Advances in Mathematics",
"publication_date": "2017-07-31",
"volume": "315",
"pages": "130-165"
},
{
"id": "authors:4evw2-jxc52",
"collection": "authors",
"collection_id": "4evw2-jxc52",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959255",
"type": "book_section",
"title": "A Note on Induced Ramsey Numbers",
"book_title": "A Journey Through Discrete Mathematics",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Dellamonica",
"given_name": "Domingos",
"clpid": "Dellamonica-Domingos"
},
{
"family_name": "La Fleur",
"given_name": "Steven",
"clpid": "La-Fleur-Steven"
},
{
"family_name": "R\u00f6dl",
"given_name": "Vojt\u011bch",
"clpid": "R\u00f6dl-Vojt\u011bch"
},
{
"family_name": "Schacht",
"given_name": "Mathias",
"clpid": "Schacht-Mathias"
}
],
"contributor": [
{
"family_name": "Loebl",
"given_name": "Martin",
"clpid": "Loebl-Martin"
},
{
"family_name": "Ne\u0161et\u0159il",
"given_name": "Jaroslav",
"clpid": "Ne\u0161et\u0159il-Jaroslav"
},
{
"family_name": "Thomas",
"given_name": "Robin",
"clpid": "Thomas-Robin"
}
],
"abstract": "The induced Ramsey number r_(ind)(F) of a k-uniform hypergraph F is the smallest natural number n for which there exists a k-uniform hypergraph G on n vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of F. We study this function, showing that r_(ind)(F) is bounded above by a reasonable power of r(F). In particular, our result implies that r_(ind)(F) \u2264 2_(2_(ct)) for any 3-uniform hypergraph F with t vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on an application of the hypergraph container method.",
"doi": "10.1007/978-3-319-44479-6_13",
"isbn": "9783319444789",
"publisher": "Springer International Publishing",
"place_of_publication": "Cham, Switzerland",
"publication_date": "2017-05-09",
"pages": "357-366"
},
{
"id": "authors:fd3nb-r4991",
"collection": "authors",
"collection_id": "fd3nb-r4991",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000547",
"type": "article",
"title": "Almost-spanning universality in random graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Ferber",
"given_name": "Asaf",
"clpid": "Ferber-Asaf"
},
{
"family_name": "Nenadov",
"given_name": "Rajko",
"clpid": "Nenadov-Rajko"
},
{
"family_name": "\u0160kori\u0107",
"given_name": "Nemanja",
"clpid": "\u0160kori\u0107-Nemanja"
}
],
"abstract": "A graph G is said to be \u210b(n, \u0394)-universal if it contains every graph on n vertices with maximum degree at most \u0394. It is known that for any \u03b5 > 0 and any natural number \u0394 there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely \u210b((1 - \u03b5)n, \u0394)-universal for p \u2265 c(log n/n)^(1/\u0394). Bypassing this natural boundary \u0394 \u2265 3, we show that for the same conclusion holds when [equation; see abstract in PDF for details].",
"doi": "10.1002/rsa.20661",
"issn": "1042-9832",
"publisher": "Wiley",
"publication": "Random Structures & Algorithms",
"publication_date": "2017-05",
"series_number": "3",
"volume": "50",
"issue": "3",
"pages": "380-393"
},
{
"id": "authors:t4b2v-c5r37",
"collection": "authors",
"collection_id": "t4b2v-c5r37",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170904052",
"type": "article",
"title": "Quasirandom Cayley graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, R\u00f6dl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.",
"doi": "10.19086/da.1294",
"issn": "2397-3129",
"publisher": "Diamond Open Access Journals",
"publication": "Discrete Analysis",
"publication_date": "2017-03-08",
"pages": "Art. No. 6"
},
{
"id": "authors:rqpp7-81c15",
"collection": "authors",
"collection_id": "rqpp7-81c15",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000651",
"type": "article",
"title": "Freiman homomorphisms on sparse random sets",
"author": [
{
"family_name": "Conlon",
"given_name": "D.",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Gowers",
"given_name": "W. T.",
"clpid": "Gowers-W-T"
}
],
"abstract": "A result of Fiz Pontiveros shows that if A is a random subset of \u2124_N where each element is chosen independently with probability N^(\u22121/2+o(1))\u2060, then with high probability every Freiman homomorphism defined on A can be extended to a Freiman homomorphism on the whole of \u2124_N\u2060. In this paper, we improve the bound to CN^(\u22122/3)(logN)^(1/3)\u2060, which is best possible up to the constant factor.",
"doi": "10.1093/qmath/haw058",
"issn": "0033-5606",
"publisher": "Oxford University Press",
"publication": "Quarterly Journal of Mathematics",
"publication_date": "2017-03",
"series_number": "1",
"volume": "68",
"issue": "1",
"pages": "275-300"
},
{
"id": "authors:g4na6-qn093",
"collection": "authors",
"collection_id": "g4na6-qn093",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000736",
"type": "article",
"title": "A Sequence of Triangle-Free Pseudorandom Graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.",
"doi": "10.1017/s0963548316000298",
"issn": "0963-5483",
"publisher": "Cambridge University Press",
"publication": "Combinatorics, Probability and Computing",
"publication_date": "2017-03",
"series_number": "2",
"volume": "26",
"issue": "2",
"pages": "195-200"
},
{
"id": "authors:v2sqk-x4v76",
"collection": "authors",
"collection_id": "v2sqk-x4v76",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000833",
"type": "article",
"title": "Ordered Ramsey numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Lee",
"given_name": "Choongbum",
"clpid": "Lee-Choongbum"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "Given a labeled graph H with vertex set {1, 2 . . ., n}, the ordered Ramsey number r < (H) is the minimum N such that every two-coloring of the edges of the complete graph on {1, 2 . . ., N} contains a copy of H with vertices appearing in the same order as in H. The ordered Ramsey number of a labeled graph H is at least the Ramsey number r(H) and the two coincide for complete graphs. However, we prove that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices. Among other results, we also prove a general upper bound on ordered Ramsey numbers which implies that there exists a constant c such that r < (H) \u2264 r(H)^(c log^(2) n) for any labeled graph H on vertex set {1, 2 . . ., n}.",
"doi": "10.1016/j.jctb.2016.06.007",
"issn": "0095-8956",
"publisher": "Elsevier",
"publication": "Journal of Combinatorial Theory, Series B",
"publication_date": "2017-01",
"volume": "122",
"pages": "353-383"
},
{
"id": "authors:csf5h-1bf02",
"collection": "authors",
"collection_id": "csf5h-1bf02",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000938",
"type": "article",
"title": "Short proofs of some extremal results II",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.",
"doi": "10.1016/j.jctb.2016.03.005",
"issn": "0095-8956",
"publisher": "Elsevier",
"publication": "Journal of Combinatorial Theory, Series B",
"publication_date": "2016-11",
"volume": "121",
"pages": "173-196"
},
{
"id": "authors:ahtz2-11b03",
"collection": "authors",
"collection_id": "ahtz2-11b03",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957976",
"type": "article",
"title": "Combinatorial theorems in sparse random sets",
"author": [
{
"family_name": "Conlon",
"given_name": "D.",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Gowers",
"given_name": "W. T.",
"clpid": "Gowers-W-T"
}
],
"abstract": "We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\u00e1n's theorem, Szemer\u00e9di's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Tur\u00e1n's theorem to the random setting by showing that for every \u03f5 > 0 and every positive integer t \u2265 3 there exists a constant C such that, if G is a random graph on n vertices where each edge is chosen independently with probability at least Cn^(\u22122/(t+1)), then, with probability tending to 1 as n tends to infinity, every subgraph of G with at least (1 \u2013 (1/(t\u22121)) + \u03f5)e(G) edges contains a copy of K_t. This is sharp up to the constant C. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Tur\u00e1n theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut, R\u00f6dl and Schacht.",
"doi": "10.4007/annals.2016.184.2.2",
"issn": "0003-486X",
"publisher": "Princeton University",
"publication": "Annals of Mathematics",
"publication_date": "2016-09",
"series_number": "2",
"volume": "184",
"issue": "2",
"pages": "367-454"
},
{
"id": "authors:985j8-0fe05",
"collection": "authors",
"collection_id": "985j8-0fe05",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001038",
"type": "article",
"title": "Monochromatic Cycle Partitions in Local Edge Colorings",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Stein",
"given_name": "Maya",
"clpid": "Stein-Maya"
}
],
"abstract": "An edge coloring of a graph is said to be an r\u2010local coloring if the edges incident to any vertex are colored with at most r colors. Generalizing a result of Bessy and Thomass\u00e9, we prove that the vertex set of any 2\u2010locally colored complete graph may be partitioned into two disjoint monochromatic cycles of different colors. Moreover, for any natural number r, we show that the vertex set of any r\u2010locally colored complete graph may be partitioned into O(r^(2) log r) disjoint monochromatic cycles. This generalizes a result of Erd\u0151s, Gy\u00e1rf\u00e1s, and Pyber.",
"doi": "10.1002/jgt.21867",
"issn": "0364-9024",
"publisher": "Wiley",
"publication": "Journal of Graph Theory",
"publication_date": "2016-02",
"series_number": "2",
"volume": "81",
"issue": "2",
"pages": "134-145"
},
{
"id": "authors:q1vhm-qth77",
"collection": "authors",
"collection_id": "q1vhm-qth77",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959158",
"type": "article",
"title": "Ramsey numbers of cubes versus cliques",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Lee",
"given_name": "Choongbum",
"clpid": "Lee-Choongbum"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. In 1983, Burr and Erd\u0151s asked whether the simple lower bound r(Q_n, K_s) \u2265 (s\u22121)(2^(n) \u2212 1) + 1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.",
"doi": "10.1007/s00493-014-3010-x",
"issn": "0209-9683",
"publisher": "Springer",
"publication": "Combinatorica",
"publication_date": "2016-02",
"series_number": "1",
"volume": "36",
"issue": "1",
"pages": "37-70"
},
{
"id": "authors:r5egw-tmm70",
"collection": "authors",
"collection_id": "r5egw-tmm70",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957531",
"type": "article",
"title": "Almost-spanning universality in random graphs (Extended abstract)",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Ferber",
"given_name": "Asaf",
"clpid": "Ferber-Asaf"
},
{
"family_name": "Nenadov",
"given_name": "Rajko",
"clpid": "Nenadov-Rajko"
},
{
"family_name": "\u0160kori\u0107",
"given_name": "Nemanja",
"clpid": "\u0160kori\u0107-Nemanja"
}
],
"abstract": "A graph G is said to be \u210b(n, \u0394)-universal if it contains every graph on n vertices with maximum degree at most \u0394. It is known that for any \u03b5 > 0 and any natural number \u0394 there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely \u210b((1 - \u03b5)n, \u0394)-universal for p \u2265 c(log n/n)^(1/\u0394). Bypassing this natural boundary \u0394 \u2265 3, we show that for the same conclusion holds when [equation; see abstract in PDF for details].",
"doi": "10.1016/j.endm.2015.06.030",
"issn": "1571-0653",
"publisher": "Elsevier",
"publication": "Electronic Notes in Discrete Mathematics",
"publication_date": "2015-11",
"volume": "49",
"pages": "203-211"
},
{
"id": "authors:mrmac-73f25",
"collection": "authors",
"collection_id": "mrmac-73f25",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959982",
"type": "article",
"title": "On the Grid Ramsey Problem and Related Questions",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Lee",
"given_name": "Choongbum",
"clpid": "Lee-Choongbum"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "The Hales-Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales-Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product K_n x K_n in r colors, then, for n sufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Shelah's proof shows that [equation; see abstract in PDF for details] suffices. More than 20 years ago, Graham, Rothschild, and Spencer asked whether this bound can be improved to a polynomial in r. We show that this is not possible by providing a superpolynomial lower bound in r. We also discuss a number of related problems.",
"doi": "10.1093/imrn/rnu190",
"issn": "1073-7928",
"publisher": "Oxford University Press",
"publication": "International Mathematics Research Notices",
"publication_date": "2015-09",
"series_number": "17",
"volume": "2015",
"issue": "17",
"pages": "8052-8084"
},
{
"id": "authors:s55mv-jtt43",
"collection": "authors",
"collection_id": "s55mv-jtt43",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190821-112738981",
"type": "video",
"title": "Rational exponents in extremal graph theory",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "[no abstract]",
"doi": "10.14288/1.0228183",
"publisher": "Banff International Research Station for Mathematical Innovation and Discovery",
"publication_date": "2015-08-25"
},
{
"id": "authors:7tq4m-xda06",
"collection": "authors",
"collection_id": "7tq4m-xda06",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959899",
"type": "book_section",
"title": "Recent developments in graph Ramsey theory",
"book_title": "Surveys in Combinatorics 2015",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"contributor": [
{
"family_name": "Czumaj",
"given_name": "Artur",
"clpid": "Czumaj-A"
},
{
"family_name": "Georgakopoulos",
"given_name": "Agelos",
"clpid": "Georgakopoulos-A"
},
{
"family_name": "Kr\u00e1l",
"given_name": "Daniel",
"clpid": "Kr\u00e1l-D"
},
{
"family_name": "Lozin",
"given_name": "Vadim",
"clpid": "Lozin-V"
},
{
"family_name": "Pikhurko",
"given_name": "Oleg",
"clpid": "Pikhurko-O"
}
],
"abstract": "Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.",
"doi": "10.1017/cbo9781316106853.003",
"isbn": "9781316106853",
"publisher": "Cambridge University Press",
"place_of_publication": "Cambridge, United Kingdom",
"publication_date": "2015-07",
"pages": "49-118"
},
{
"id": "authors:fecwr-jw093",
"collection": "authors",
"collection_id": "fecwr-jw093",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959067",
"type": "article",
"title": "A relative Szemer\u00e9di theorem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\u00e9di theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemer\u00e9di theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemer\u00e9di theorem for k-term arithmetic progressions in pseudorandom subsets of \u2124_N of density N^(\u2212ck). The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemer\u00e9di theorem.",
"doi": "10.1007/s00039-015-0324-9",
"issn": "1016-443X",
"publisher": "Springer",
"publication": "Geometric and Functional Analysis",
"publication_date": "2015-06",
"series_number": "3",
"volume": "25",
"issue": "3",
"pages": "733-762"
},
{
"id": "authors:p1qtp-rka12",
"collection": "authors",
"collection_id": "p1qtp-rka12",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958151",
"type": "article",
"title": "Distinct Volume Subsets",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Gasarch",
"given_name": "William",
"clpid": "Gasarch-W"
},
{
"family_name": "Harris",
"given_name": "David G.",
"clpid": "Harris-D-G"
},
{
"family_name": "Ulrich",
"given_name": "Douglas",
"clpid": "Ulrich-D"
},
{
"family_name": "Zbarsky",
"given_name": "Samuel",
"clpid": "Zbarsky-S"
}
],
"abstract": "Suppose that a and d are positive integers with a \u2265 2. Let h_(a,d)(n) be the largest integer t such that any set of n points in \u211d^d contains a subset of t points for which all the nonzero volumes of the [equaton; see abstract in PDF for details] subsets of order a are distinct. Beginning with Erd\u0151s in 1957, the function h_(2,d)(n) has been closely studied and is known to be at least a power of n. We improve the best known bound for h_(2,d)(n) and show that h_(a,d)(n) is at least a power of n for all a and d.",
"doi": "10.1137/140954519",
"issn": "0895-4801",
"publisher": "Society for Industrial and Applied Mathematics",
"publication": "SIAM Journal on Discrete Mathematics",
"publication_date": "2015-03-11",
"series_number": "1",
"volume": "29",
"issue": "1",
"pages": "472-480"
},
{
"id": "authors:mrfhv-pps81",
"collection": "authors",
"collection_id": "mrfhv-pps81",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959803",
"type": "article",
"title": "The Erd\u0151s-Gy\u00e1rf\u00e1s problem on generalized Ramsey numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Lee",
"given_name": "Choongbum",
"clpid": "Lee-Choongbum"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "Fix positive integers p and q with [equation; see abstract in PDF for details]. An edge coloring of the complete graph K_n is said to be a (p,q)-coloring if every K_p receives at least q different colors. The function f(n,p,q) is the minimum number of colors that are needed for K_n to have a (p,q)-coloring. This function was introduced about 40 years ago, but Erd\u0151s and Gy\u00e1rf\u00e1s were the first to study the function in a systematic way. They proved that f(n,p,p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n,p,q) is subpolynomial in n. We prove that the answer is p - 1.",
"doi": "10.1112/plms/pdu049",
"issn": "0024-6115",
"publisher": "London Mathematical Society",
"publication": "Proceedings of the London Mathematical Society",
"publication_date": "2015-01",
"series_number": "1",
"volume": "110",
"issue": "1",
"pages": "1-18"
},
{
"id": "authors:2xrg4-tje16",
"collection": "authors",
"collection_id": "2xrg4-tje16",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001129",
"type": "article",
"title": "Cycle packing",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "In the 1960s, Erd\u0151s and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(nlogn) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(nloglogn) cycles and edges suffice. We also prove the Erd\u0151s\u2010Gallai conjecture for random graphs and for graphs with linear minimum degree.",
"doi": "10.1002/rsa.20574",
"issn": "1042-9832",
"publisher": "Wiley",
"publication": "Random Structures & Algorithms",
"publication_date": "2014-12",
"series_number": "4",
"volume": "45",
"issue": "4",
"pages": "608-626"
},
{
"id": "authors:pfh4w-3qv46",
"collection": "authors",
"collection_id": "pfh4w-3qv46",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958948",
"type": "article",
"title": "On the K\u0141R conjecture in random graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "D.",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Gowers",
"given_name": "W. T.",
"clpid": "Gowers-W-T"
},
{
"family_name": "Samotij",
"given_name": "W.",
"clpid": "Samotij-W"
},
{
"family_name": "Schacht",
"given_name": "M.",
"clpid": "Schacht-M"
}
],
"abstract": "The K\u0141R conjecture of Kohayakawa, \u0141uczak, and R\u00f6dl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G_(n, p), for sufficiently large p := p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and R\u00f6dl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.",
"doi": "10.1007/s11856-014-1120-1",
"issn": "0021-2172",
"publisher": "Springer",
"publication": "Israel Journal of Mathematics",
"publication_date": "2014-10",
"series_number": "1",
"volume": "203",
"issue": "1",
"pages": "535-580"
},
{
"id": "authors:0hxg6-rgb55",
"collection": "authors",
"collection_id": "0hxg6-rgb55",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000075",
"type": "article",
"title": "Ramsey-type results for semi-algebraic relations",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Pach",
"given_name": "J\u00e1nos",
"clpid": "Pach-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
},
{
"family_name": "Suk",
"given_name": "Andrew",
"clpid": "Suk-Andrew"
}
],
"abstract": "A k-ary semi-algebraic relation E on \u211d_d is a subset of \u211d_(kd), the set of k-tuples of points in \u211d_(d), which is determined by a finite number of polynomial inequalities in kd real variables. The description complexity of such a relation is at most t if d, k \u2264 t and the number of polynomials and their degrees are all bounded by t. A set A \u2282 \u211d_d is called homogeneous if all or none of the k-tuples from A satisfy E. A large number of geometric Ramsey-type problems and results can be formulated as questions about finding large homogeneous subsets of sets in \u211d_d equipped with semi-algebraic relations.\n\nIn this paper, we study Ramsey numbers for k-ary semi-algebraic relations of bounded complexity and give matching upper and lower bounds, showing that they grow as a tower of height k \u2212 1. This improves upon a direct application of Ramsey's theorem by one exponential and extends a result of Alon, Pach, Pinchasi, Radoi\u010di\u0107, and Sharir, who proved this for k = 2. We apply our results to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.",
"doi": "10.1090/s0002-9947-2014-06179-5",
"issn": "0002-9947",
"publisher": "American Mathematical Society",
"publication": "Transactions of the American Mathematical Society",
"publication_date": "2014-09",
"series_number": "9",
"volume": "366",
"issue": "9",
"pages": "5043-5065"
},
{
"id": "authors:2a4a3-ndy66",
"collection": "authors",
"collection_id": "2a4a3-ndy66",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957263",
"type": "article",
"title": "Extremal results in sparse pseudorandom graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "Szemer\u00e9di's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\u00f6dl proved an analogue of Szemer\u00e9di's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\u00e9di's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs.\n\nThe main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\u00f6dl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\u0151s\u2013Stone\u2013Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.",
"doi": "10.1016/j.aim.2013.12.004",
"issn": "0001-8708",
"publisher": "Elsevier",
"publication": "Advances in Mathematics",
"publication_date": "2014-05-01",
"volume": "256",
"pages": "206-290"
},
{
"id": "authors:ew3yz-n7p38",
"collection": "authors",
"collection_id": "ew3yz-n7p38",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162956937",
"type": "article",
"title": "Short Proofs of Some Extremal Results",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.",
"doi": "10.1017/s0963548313000448",
"issn": "0963-5483",
"publisher": "Cambridge University Press",
"publication": "Combinatorics, Probability and Computing",
"publication_date": "2014-01",
"series_number": "1",
"volume": "23",
"issue": "1",
"pages": "8-28"
},
{
"id": "authors:8zjkv-zvw34",
"collection": "authors",
"collection_id": "8zjkv-zvw34",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001431",
"type": "article",
"title": "The Green-Tao theorem: an exposition",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Zhao",
"given_name": "Yufei",
"clpid": "Zhao-Yufei"
}
],
"abstract": "The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.",
"doi": "10.4171/emss/6",
"issn": "2308-2151",
"publisher": "European Mathematical Society",
"publication": "EMS Surveys in Mathematical Sciences",
"publication_date": "2014",
"series_number": "2",
"volume": "1",
"issue": "2",
"pages": "249-282"
},
{
"id": "authors:6xzh8-1ca79",
"collection": "authors",
"collection_id": "6xzh8-1ca79",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190820-164721413",
"type": "book_section",
"title": "Combinatorial theorems relative to a random set",
"book_title": "Proceedings of the International Congress of Mathematicians",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"contributor": [
{
"family_name": "Jang",
"given_name": "Sun Young",
"clpid": "Jang-Sun-Young"
},
{
"family_name": "Kim",
"given_name": "Young Rock",
"clpid": "Kim-Young-Rock"
},
{
"family_name": "Lee",
"given_name": "Dae-Woong",
"clpid": "Lee-Dae-Woong"
},
{
"family_name": "Yie",
"given_name": "Ikkwon",
"clpid": "Yie-Ikkwon"
}
],
"abstract": "We describe recent advances in the study of random analogues of combinatorial theorems.",
"doi": "10.48550/arXiv.1404.3324",
"isbn": "9788961058070",
"publisher": "Kyung Moon Sa Co. Ltd.",
"place_of_publication": "Seoul, Korea",
"publication_date": "2014",
"pages": "303-327"
},
{
"id": "authors:ryxaq-f1122",
"collection": "authors",
"collection_id": "ryxaq-f1122",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959551",
"type": "article",
"title": "Two extensions of Ramsey's theorem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "Ramsey's theorem, in the version of Erd\u0151s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2, . . ., n} contains a monochromatic clique of order (1/2)log n. In this article, we consider two well-studied extensions of Ramsey's theorem. Improving a result of R\u00f6dl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, . . ., n} contains a monochromatic clique S for which the sum of 1/log i over all vertices i \u2208 S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erd\u0151s from 1981. Motivated by a problem in model theory, V\u00e4\u00e4n\u00e4nen asked whether for every k there is an n such that the following holds: for every permutation \u03c0 of 1, . . ., k\u22121, every 2-coloring of the edges of the complete graph on {1, 2, . . ., n} contains a monochromatic clique a_1 < \u2022 \u2022 \u2022 < a_k with\n\na_(\u03c0(1)+1) \u2212 a_(\u03c0(1)) > a_\u03c0((2)+1) \u2212 a_(\u03c0(2)) > \u2022 \u2022 \u2022 > a_(\u03c0(k\u22121)+1) \u2212 a_(\u03c0(k\u22121)).\n\nThat is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erd\u0151s, Hajnal, and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.",
"doi": "10.1215/00127094-2382566",
"issn": "0012-7094",
"publisher": "Duke University Press",
"publication": "Duke Mathematical Journal",
"publication_date": "2013-11-28",
"series_number": "15",
"volume": "162",
"issue": "15",
"pages": "2903-2927"
},
{
"id": "authors:kk69w-7xs20",
"collection": "authors",
"collection_id": "kk69w-7xs20",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001516",
"type": "book_section",
"title": "Graph removal lemmas",
"book_title": "Surveys in Combinatorics 2013",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
}
],
"contributor": [
{
"family_name": "Blackburn",
"given_name": "Simon R.",
"clpid": "Blackburn-S-R"
},
{
"family_name": "Gerke",
"given_name": "Stefanie",
"clpid": "Gerke-S"
},
{
"family_name": "Wildon",
"given_name": "Mark",
"clpid": "Wildon-M"
}
],
"abstract": "The graph removal lemma states that any graph on n vertices with o(n^h) copies of a fixed graph H on h vertices may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.",
"doi": "10.1017/cbo9781139506748.002",
"isbn": "9781139506748",
"publisher": "Cambridge University Press",
"place_of_publication": "Cambridge, United Kingdom",
"publication_date": "2013-07",
"pages": "1-50"
},
{
"id": "authors:sp7g8-vxr43",
"collection": "authors",
"collection_id": "sp7g8-vxr43",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-170836067",
"type": "article",
"title": "The Ramsey number of dense graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "The Ramsey number r(H) of a graph H is the smallest number n such that, in any two-colouring of the edges of K_n, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t vertices and density \u03c1, proving that r(H) \u2264 2^(c\u221a(\u03c1)log(2/\u03c1)t). We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree \u03c1t and the Ramsey number of random graphs in G(t, \u03c1), that is, graphs on t vertices where each edge has been chosen independently with probability \u03c1.",
"doi": "10.1112/blms/bds097",
"issn": "0024-6093",
"publisher": "London Mathematical Society",
"publication": "Bulletin of the London Mathematical Society",
"publication_date": "2013-06",
"series_number": "3",
"volume": "45",
"issue": "3",
"pages": "483-496"
},
{
"id": "authors:5hz8p-e8k15",
"collection": "authors",
"collection_id": "5hz8p-e8k15",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957165",
"type": "article",
"title": "An improved bound for the stepping-up lemma",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "The partition relation N \u2192 (n)^(k)_(\u2113) means that whenever the k-tuples of an N-element set are \u2113-colored, there is a monochromatic set of size n, where a set is called monochromatic if all its k-tuples have the same color. The logical negation of N \u2192 (n)^(k)_(\u2113) is written as N / \u2192 (n)^(k)_(\u2113). An ingenious construction of Erd\u0151s and Hajnal known as the stepping-up lemma gives a negative partition relation for higher uniformity from one of lower uniformity, effectively gaining an exponential in each application. Namely, if \u2113 \u2265 2, k \u2265 3, and N / \u2192 (n)^(k)_(\u2113), then 2^N / \u2192 (2n + k - 4)^(k+1)_(\u2113). In this paper we give an improved construction for k \u2265 4. We introduce a general class of colorings which extends the framework of Erd\u0151s and Hajnal and can be used to establish negative partition relations. We show that if \u2113 \u2265 2, k \u2265 4 and N / \u2192 (n)^(k)_(\u2113), then 2^N / \u2192 (n + 3)^(k+1)_(\u2113). If also k is odd or \u2113 \u2265 3, then we get the better bound 2^N / \u2192 (n + 2)^(k+1)_(\u2113). This improved bound gives a coloring of the k-tuples whose largest monochromatic set is a factor \u03a9(2^k) smaller than that given by the original version of the stepping-up lemma. We give several applications of our result to lower bounds on hypergraph Ramsey numbers. In particular, for fixed \u2113 \u2265 4 we determine up to an absolute constant factor (which is independent of k) the size of the largest guaranteed monochromatic set in an \u2113-coloring of the k-tuples of an N-set.",
"doi": "10.1016/j.dam.2010.10.013",
"issn": "0166-218X",
"publisher": "Elsevier",
"publication": "Discrete Applied Mathematics",
"publication_date": "2013-06",
"series_number": "9",
"volume": "161",
"issue": "9",
"pages": "1191-1196"
},
{
"id": "authors:5dhpf-yek21",
"collection": "authors",
"collection_id": "5dhpf-yek21",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959355",
"type": "article",
"title": "Bounds for graph regularity and removal lemmas",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
}
],
"abstract": "We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^(2)/log^* k pairs of parts which are not \u03f5-regular, where c, \u03f5 > 0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\u00e9di's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \u03f5 may require as many as 2^\u03a9(\u03f5^(\u22122)) parts. This is tight up to the implied constant and solves a problem studied by Lov\u00e1sz and Szegedy.",
"doi": "10.1007/s00039-012-0171-x",
"issn": "1016-443X",
"publisher": "Springer",
"publication": "Geometric and Functional Analysis",
"publication_date": "2012-10",
"series_number": "5",
"volume": "22",
"issue": "5",
"pages": "1191-1256"
},
{
"id": "authors:69ssa-jy057",
"collection": "authors",
"collection_id": "69ssa-jy057",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957444",
"type": "article",
"title": "Erd\u0151s-Hajnal-type theorems in hypergraphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "The Erd\u0151s\u2013Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^\u03b4(H), where \u03b4(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that an H-free graph must contain a complete or empty bipartite graph with parts of polynomial size.\n\nWe prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is \u210b-free, for any given \u210b, then it must contain a complete or empty tripartite subgraph with parts of order c(log n)^\u00bd + \u03b4(\u210b), where \u03b4(\u210b) > 0 depends only on \u210b. This improves on the bound of c(log n)^\u00bd, which holds in all 3-uniform hypergraphs, and, up to the value of the constant \u03b4(\u210b), is best possible.\n\nWe also prove that, for k \u2265 4, no analogue of the standard Erd\u0151s\u2013Hajnal conjecture can hold in k-uniform hypergraphs. That is, there are k-uniform hypergraphs \u210b and sequences of \u210b-free hypergraphs which do not contain cliques or independent sets of size appreciably larger than one would normally expect.",
"doi": "10.1016/j.jctb.2012.05.005",
"issn": "0095-8956",
"publisher": "Elsevier",
"publication": "Journal of Combinatorial Theory, Series B",
"publication_date": "2012-09",
"series_number": "5",
"volume": "102",
"issue": "5",
"pages": "1142-1154"
},
{
"id": "authors:s358b-c8d04",
"collection": "authors",
"collection_id": "s358b-c8d04",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958859",
"type": "article",
"title": "On two problems in graph Ramsey theory",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. \n\nThe Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph K_N contains a monochromatic copy of H. A famous result of Chv\u00e1tal, R\u00f6dl, Szemer\u00e9di and Trotter states that there exists a constant c(\u0394) such that r(H) \u2264 c(\u0394)n for every graph H with n vertices and maximum degree \u0394. The important open question is to determine the constant c(\u0394). The best results, both due to Graham, R\u00f6dl and Ruci\u0144ski, state that there are positive constants c and c' such that 2^(c'\u0394) \u2264 c(\u0394) \u2264 c^(\u0394log^(2)\u0394). We improve this upper bound, showing that there is a constant c for which c(\u0394) \u2264 2^(c\u0394log\u0394).",
"doi": "10.1007/s00493-012-2710-3",
"issn": "0209-9683",
"publisher": "Springer",
"publication": "Combinatorica",
"publication_date": "2012-05",
"series_number": "5",
"volume": "32",
"issue": "5",
"pages": "513-535"
},
{
"id": "authors:60yc2-spk66",
"collection": "authors",
"collection_id": "60yc2-spk66",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958767",
"type": "article",
"title": "On the Ramsey multiplicity of complete graphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We show that, for n large, there must exist at least \n\n(n^t)/(C^((1+o(1))t^2)) \n\nmonochromatic K_(t)s in any two-colouring of the edges of K_n, where C \u2248 2.18 is an explicitly defined constant. The old lower bound, due to Erd\u0151s [2], and based upon the standard bounds for Ramsey's theorem, is \n\n(n^t)/(4^((1+o(1))t^2)).",
"doi": "10.1007/s00493-012-2465-x",
"issn": "0209-9683",
"publisher": "Springer",
"publication": "Combinatorica",
"publication_date": "2012-03",
"series_number": "2",
"volume": "32",
"issue": "2",
"pages": "171-186"
},
{
"id": "authors:7zy71-zzw53",
"collection": "authors",
"collection_id": "7zy71-zzw53",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001217",
"type": "article",
"title": "Weak quasi-randomness for uniform hypergraphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "H\u00e0n",
"given_name": "Hi\u00eap",
"clpid": "H\u00e0n-Hi\u00eap"
},
{
"family_name": "Person",
"given_name": "Yury",
"clpid": "Person-Y"
},
{
"family_name": "Schacht",
"given_name": "Mathias",
"clpid": "Schacht-M"
}
],
"abstract": "We study quasi\u2010random properties of k\u2010uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung\u2010Graham\u2010Wilson theorem for quasi\u2010random graphs.\n\nMoreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k\u2010dimensional hypercube. We will show that the pair of graphs (K_(k),M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi\u2010random (in the sense of the Chung\u2010Graham\u2010Wilson theorem) with density close to d.",
"doi": "10.1002/rsa.20389",
"issn": "1042-9832",
"publisher": "Wiley",
"publication": "Random Structures & Algorithms",
"publication_date": "2012-01",
"series_number": "1",
"volume": "40",
"issue": "1",
"pages": "1-38"
},
{
"id": "authors:rgb7g-cgs47",
"collection": "authors",
"collection_id": "rgb7g-cgs47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958247",
"type": "article",
"title": "Inevitable randomness in discrete mathematics [Book Review]",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "Complex systems arise naturally throughout mathematics, computer science, economics and the physical sciences. However, when faced with such intricate systems, we are often powerless to understand their behaviour. Without simplifying assumptions, such as that of the rational agent in economics, the size of the space of possibilities and its apparent lack of order can become overwhelming.\n\nIn this book, J\u00f3zsef Beck explores this issue, suggesting that discrete systems which are not simple should always behave in a random-like fashion, even when different parts of the system do not behave independently.\n\nThe central focus of the book is, to use the author's own term, the following 'vague' metaconjecture.",
"doi": "10.1112/blms/bdr063",
"issn": "0024-6093",
"publisher": "London Mathematical Society",
"publication": "Bulletin of the London Mathematical Society",
"publication_date": "2011-10",
"series_number": "5",
"volume": "43",
"issue": "5",
"pages": "1021-1023"
},
{
"id": "authors:jmby0-yas06",
"collection": "authors",
"collection_id": "jmby0-yas06",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958677",
"type": "article",
"title": "Large almost monochromatic subsets in hypergraphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We show that for all \u2113 and \u03b5 > 0 there is a constant c = c(\u2113, \u03b5) > 0 such that every \u2113-coloring of the triples of an N-element set contains a subset S of size c\u221a(log N) such that at least 1 \u2212 \u03b5 fraction of the triples of S have the same color. This result is tight up to the constant c and answers an open question of Erd\u0151s and Hajnal from 1989 on discrepancy in hypergraphs. For \u2113 \u2265 4 colors, it is known that there is an \u2113-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only \u0398(log log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an \u2113-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the \u2113-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erd\u0151s and Hajnal.",
"doi": "10.1007/s11856-011-0016-6",
"issn": "0021-2172",
"publisher": "Springer",
"publication": "Israel Journal of Mathematics",
"publication_date": "2011-01",
"series_number": "1",
"volume": "181",
"issue": "1",
"pages": "423-432"
},
{
"id": "authors:z6qfe-0dd58",
"collection": "authors",
"collection_id": "z6qfe-0dd58",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958544",
"type": "article",
"title": "An Approximate Version of Sidorenko's Conjecture",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "A beautiful conjecture of Erd\u0151s-Simonovits and Sidorenko states that, if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.",
"doi": "10.1007/s00039-010-0097-0",
"issn": "1016-443X",
"publisher": "Springer",
"publication": "Geometric and Functional Analysis",
"publication_date": "2010-12",
"series_number": "6",
"volume": "20",
"issue": "6",
"pages": "1354-1366"
},
{
"id": "authors:pw80h-jdy02",
"collection": "authors",
"collection_id": "pw80h-jdy02",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190819-163059223",
"type": "article",
"title": "An Extremal Theorem in the Hypercube",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "The hypercube Q_n is the graph whose vertex set is {0, 1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_(n), H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_(n), H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_(n), C_(2t)) = o(e(Q_n)) for all t \u2265 4 other than 5.",
"doi": "10.48550/arXiv.1005.0582",
"issn": "1077-8926",
"publisher": "Electronic Journal of Combinatorics",
"publication": "Electronic Journal of Combinatorics",
"publication_date": "2010-08-09",
"volume": "17",
"pages": "Art. No. R111"
},
{
"id": "authors:tmnj9-7nm31",
"collection": "authors",
"collection_id": "tmnj9-7nm31",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000262",
"type": "article",
"title": "Hypergraph Ramsey numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "The Ramsey number r_(k)(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_(k)(s, n) for k \u2265 3 and s fixed. In particular, we show that \n\nr_(3)(s, n) \u2264 2^(n^(s-2)log n), \n\nwhich improves by a factor of n^(s-2)/polylog n the exponent of the previous upper bound of Erd\u0151s and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant c > 0 such that \n\nr_(3)(s, n) \u2265 2^(csn log((n/s)+1)) \n\nfor all 4 \u2264 s \u2264 n. For constant s, this gives the first superexponential lower bound for r_(3)(s, n), answering an open question posed by Erd\u0151s and Hajnal in 1972. Next, we consider the 3-color Ramsey number r_(3)(n, n, n), which is the minimum N such that every 3-coloring of the triples of an N-element set contains a monochromatic set of size n. Improving another old result of Erd\u0151s and Hajnal, we show that \n\nr_(3)(n, n, n) \u2265 2^(n^(c log n)). \n\nFinally, we make some progress on related hypergraph Ramsey-type problems.",
"doi": "10.1090/s0894-0347-09-00645-6",
"issn": "0894-0347",
"publisher": "American Mathematical Society",
"publication": "Journal of the American Mathematical Society",
"publication_date": "2010-01",
"series_number": "1",
"volume": "23",
"issue": "1",
"pages": "247-266"
},
{
"id": "authors:fff0b-1g810",
"collection": "authors",
"collection_id": "fff0b-1g810",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958058",
"type": "article",
"title": "On-line Ramsey Numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colors them in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number \u02dcr(G) of G. Our main result, relating to the conjecture that [equation; see abstract in PDF for details], is that there exists a constant c > 1 such that [equation; see abstract in PDF for details] for infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a positive constant c such that [equation; see abstract in PDF for details]. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph K_(t,t).",
"doi": "10.1137/090749220",
"issn": "0895-4801",
"publisher": "Society for Industrial and Applied Mathematics",
"publication": "SIAM Journal on Discrete Mathematics",
"publication_date": "2009-12-11",
"series_number": "4",
"volume": "23",
"issue": "4",
"pages": "1954-1963"
},
{
"id": "authors:41g6c-zjc92",
"collection": "authors",
"collection_id": "41g6c-zjc92",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959718",
"type": "article",
"title": "Hypergraph Packing and Sparse Bipartite Ramsey Numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We prove that there exists a constant c such that, for any integer \u0394, the Ramsey number of a bipartite graph on n vertices with maximum degree \u0394 is less than 2^(c\u0394)n. A probabilistic argument due to Graham, R\u00f6dl and Ruci\u0144ski implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of R\u00f6dl, Ruci\u0144ski and Taraz.",
"doi": "10.1017/s0963548309990174",
"issn": "0963-5483",
"publisher": "Cambridge University Press",
"publication": "Combinatorics, Probability and Computing",
"publication_date": "2009-11",
"series_number": "6",
"volume": "18",
"issue": "6",
"pages": "913-923"
},
{
"id": "authors:vjwec-s3c53",
"collection": "authors",
"collection_id": "vjwec-s3c53",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957624",
"type": "article",
"title": "A new upper bound for diagonal Ramsey numbers",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant C such that \n\nr(k + 1, k + 1) \u2264 k^[-C log k/(log log k (2k_k).",
"doi": "10.4007/annals.2009.170.941",
"issn": "0003-486X",
"publisher": "Princeton University",
"publication": "Annals of Mathematics",
"publication_date": "2009-09",
"series_number": "2",
"volume": "170",
"issue": "2",
"pages": "941-960"
},
{
"id": "authors:4mrey-82r45",
"collection": "authors",
"collection_id": "4mrey-82r45",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001597",
"type": "article",
"title": "Ramsey numbers of sparse hypergraphs",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Fox",
"given_name": "Jacob",
"clpid": "Fox-J"
},
{
"family_name": "Sudakov",
"given_name": "Benny",
"clpid": "Sudakov-B"
}
],
"abstract": "We give a short proof that any k\u2010uniform hypergraph H on n vertices with bounded degree \u0394 has Ramsey number at most c(\u0394,k)n, for an appropriate constant c(\u0394,k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self\u2010contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and R\u00f6dl. Moreover, our method demonstrates that, for k \u2265 4,\n\n[equation; see abstract in PDF for details],\n\nwhere the tower is of height k and the constant c depends on k. It significantly improves on the Ackermann\u2010type upper bound that arises from the regularity proofs, and we present a construction which shows that, at least in certain cases, this bound is not far from best possible. Our methods also allows us to prove quite sharp results on the Ramsey number of hypergraphs with at most m edges.",
"doi": "10.1002/rsa.20260",
"issn": "1042-9832",
"publisher": "Wiley",
"publication": "Random Structures & Algorithms",
"publication_date": "2009-08",
"series_number": "1",
"volume": "35",
"issue": "1",
"pages": "1-14"
},
{
"id": "authors:m4gwj-46284",
"collection": "authors",
"collection_id": "m4gwj-46284",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001317",
"type": "article",
"title": "A new upper bound for the bipartite Ramsey problem",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We consider the following question: how large does n have to be to guarantee that in any two\u2010coloring of the edges of the complete graph K_(n,n) there is a monochromatic K_(k,k)? In the late 1970s, Irving showed that it was sufficient, for k large, that n\u2009\u2265 2^(k\u2009\u2212 1) (k\u2009\u2212 1) \u2212 1. Here we improve upon this bound, showing that it is sufficient to take\n\nn \u2265 (1 + o(1))2^(k+1) log k,\n\nwhere the log is taken to the base 2.",
"doi": "10.1002/jgt.20317",
"issn": "0364-9024",
"publisher": "Wiley",
"publication": "Journal of Graph Theory",
"publication_date": "2008-08",
"series_number": "4",
"volume": "58",
"issue": "4",
"pages": "351-356"
},
{
"id": "authors:gn36h-40570",
"collection": "authors",
"collection_id": "gn36h-40570",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958451",
"type": "article",
"title": "On the Existence of Rainbow 4-Term Arithmetic Progressions",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Jungi\u0107",
"given_name": "Veselin",
"clpid": "Jungi\u0107-Veselin"
},
{
"family_name": "Radoi\u010di\u0107",
"given_name": "Rado\u0161",
"clpid": "Radoi\u010di\u0107-Rado\u0161"
}
],
"abstract": "For infinitely many natural numbers n, we construct 4-colorings of [n] = {1, 2, . . ., n}, with equinumerous color classes, that contain no 4-term arithmetic progression whose elements are colored in distinct colors. This result solves an open problem of Jungi\u0107 et al. (Comb Probab Comput 12:599\u2013620, 2003) Axenovich and Fon-der-Flaass (Electron J Comb 11:R1, 2004).",
"doi": "10.1007/s00373-007-0723-2",
"issn": "0911-0119",
"publisher": "Springer",
"publication": "Graphs and Combinatorics",
"publication_date": "2007-06",
"series_number": "3",
"volume": "23",
"issue": "3",
"pages": "249-254"
},
{
"id": "authors:dgwdk-sq532",
"collection": "authors",
"collection_id": "dgwdk-sq532",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957077",
"type": "article",
"title": "Rainbow solutions of linear equations over \u2124_p",
"author": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"abstract": "We prove that if the group \u2124_p, with p a prime, is coloured with k \u2265 4 different colours such that each colour appears at least k times, then for any a_1, . . . a_k, b in \u2124_p with not all the a_i being equal, we may solve the equation A_(1)x_(1) + \u2022 \u2022 \u2022 + a_(k)x_(k) = b so that each of the variables is chosen in a different colour class. This generalises a similar result concerning three colour classes due to Jungi\u0107, Licht, Mahdian, Ne\u0161et\u0159il and Radoi\u010di\u0107.\n\nIn the course of our proof we classify, with some size caveats, the sets in \u2124_p which satisfy the inequality | A_1 + \u2022 \u2022 \u2022 + A_n | \u2264 | A_1 | + \u2022 \u2022 \u2022 + | A_1 |. This is a generalisation of an inverse theorem due to Hamidoune and R\u00f8dseth concerning the case n = 2.",
"doi": "10.1016/j.disc.2006.03.070",
"issn": "0012-365X",
"publisher": "Elsevier",
"publication": "Discrete Mathematics",
"publication_date": "2006-09-06",
"series_number": "17",
"volume": "306",
"issue": "17",
"pages": "2056-2063"
}
]