CaltechAUTHORS: Monograph
https://feeds.library.caltech.edu/people/Conlon-David/monograph.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 20 Sep 2024 12:55:42 -0700Hypergraph cuts above the average
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170914451
Year: 2019
DOI: 10.48550/arXiv.1803.08462
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 + Ω(√m), 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 Ω(√m) 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 ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170914451Graphs with few paths of prescribed length between any two vertices
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170853333
Year: 2019
DOI: 10.48550/arXiv.1411.0856
We use a variant of Bukh's random algebraic method to show that for every natural number k ≥ 2 there exists a natural number ℓ such that, for every n, there is a graph with n vertices and Ω_(k)(n^(1 + 1/k)) edges with at most ℓ 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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170853333Linear forms from the Gowers uniformity norm
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170846378
Year: 2019
DOI: 10.48550/arXiv.1305.5565
This is a companion note to our paper 'A relative Szemerédi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemerédi 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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170846378Hypergraph expanders from Cayley graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170907486
Year: 2019
DOI: 10.48550/arXiv.1709.10006
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 ℤ^(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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170907486Independent arithmetic progressions
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170939635
Year: 2019
DOI: 10.48550/arXiv.1901.05084
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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170939635A note on lower bounds for hypergraph Ramsey numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170832267
Year: 2019
DOI: 10.48550/arXiv.0711.5004
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that
r_(3)(l, l, l) ≥ 2^(l^(c log log l)).
The old bound, due to Erdős and Hajnal, was
r_(3)(l, l, l) ≥ 2^(cl^(2) log^(2) l).https://resolver.caltech.edu/CaltechAUTHORS:20190819-170832267Large subgraphs without complete bipartite graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170849884
Year: 2019
DOI: 10.48550/arXiv.1401.6711
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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170849884Online Ramsey Numbers and the Subgraph Query Problem
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170917920
Year: 2019
DOI: 10.48550/arXiv.1806.09726
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
[equation; see abstract in PDF for details]
for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement
[equation; see abstract in PDF for details]
for the off-diagonal case, where m ≥ 3 is fixed and n → ∞. 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 ≥ 4.
In 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 Erdős-Rényi 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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170917920On the extremal number of subdivisions
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170921338
Year: 2019
DOI: 10.48550/arXiv.1807.05008
One of the cornerstones of extremal graph theory is a result of Füredi, 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 Füredi'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 δ such that every graph with n vertices and Cn^(3/2) - δ) edges contains a copy of H. This answers a question of Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170921338Hedgehogs are not colour blind
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170900578
Year: 2019
DOI: 10.48550/arXiv.1511.00563
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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170900578Sidorenko's conjecture for blow-ups
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170928301
Year: 2019
DOI: 10.48550/arXiv.1809.01259
A celebrated conjecture of Sidorenko and Erdős-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.
Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A ∪ 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 ∪ 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.
Another 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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170928301Sidorenko's conjecture for a class of graphs: an exposition
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170842963
Year: 2019
DOI: 10.48550/arXiv.1209.0184
A famous conjecture of Sidorenko and Erdős-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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170842963Intervals in the Hales-Jewett theorem
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170911002
Year: 2019
DOI: 10.48550/arXiv.1801.08919
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 ⊆ [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.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170911002The Ramsey number of books
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800
Year: 2019
DOI: 10.48550/arXiv.1808.03157
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 Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.https://resolver.caltech.edu/CaltechAUTHORS:20190819-170924800Sidorenko's conjecture for higher tree decompositions
https://resolver.caltech.edu/CaltechAUTHORS:20190820-161459631
Year: 2019
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.https://resolver.caltech.edu/CaltechAUTHORS:20190820-161459631A New Bound for the Brown-Erdős-Sós Problem
https://resolver.caltech.edu/CaltechAUTHORS:20200915-144809437
Year: 2020
DOI: 10.48550/arXiv.1912.08834
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≥3, what is the smallest integer d=d(e) so that f(n,e+d,e)=o(n²)? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e)=3 for every e≥3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f(n,e+2+⌊log₂e⌋,e)=o(n²). 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 Sárközy--Selkow bound, showing that
f(n,e+O(loge/logloge),e)=o(n²).https://resolver.caltech.edu/CaltechAUTHORS:20200915-144809437Which graphs can be counted in C₄-free graphs?
https://resolver.caltech.edu/CaltechAUTHORS:20230606-676918000.2
Year: 2021
For which graphs F is there a sparse F-counting lemma in C₄-free graphs? We are interested in identifying graphs F with the property that, roughly speaking, if G is an n-vertex C₄-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 ε-regular approximation of G. In recent work, motivated by applications in extremal and additive combinatorics, we showed that C₅ has this property. Here we construct a family of graphs with the property.https://resolver.caltech.edu/CaltechAUTHORS:20230606-676918000.2