Article records
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 17:55:37 +0000Rainbow solutions of linear equations over ℤ_p
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957077
Authors: Conlon, David
Year: 2006
DOI: 10.1016/j.disc.2006.03.070
We prove that if the group ℤ_p, with p a prime, is coloured with k ≥ 4 different colours such that each colour appears at least k times, then for any a_1, . . . a_k, b in ℤ_p with not all the a_i being equal, we may solve the equation A_(1)x_(1) + • • • + 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 Jungić, Licht, Mahdian, Nešetřil and Radoičić.
In the course of our proof we classify, with some size caveats, the sets in ℤ_p which satisfy the inequality | A_1 + • • • + A_n | ≤ | A_1 | + • • • + | A_1 |. This is a generalisation of an inverse theorem due to Hamidoune and Rødseth concerning the case n = 2.https://authors.library.caltech.edu/records/dgwdk-sq532On the Existence of Rainbow 4-Term Arithmetic Progressions
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958451
Authors: Conlon, David; Jungić, Veselin; Radoičić, Radoš
Year: 2007
DOI: 10.1007/s00373-007-0723-2
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 Jungić et al. (Comb Probab Comput 12:599–620, 2003) Axenovich and Fon-der-Flaass (Electron J Comb 11:R1, 2004).https://authors.library.caltech.edu/records/gn36h-40570A new upper bound for the bipartite Ramsey problem
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001317
Authors: Conlon, David
Year: 2008
DOI: 10.1002/jgt.20317
We consider the following question: how large does n have to be to guarantee that in any two‐coloring 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 ≥ 2^(k − 1) (k − 1) − 1. Here we improve upon this bound, showing that it is sufficient to take
n ≥ (1 + o(1))2^(k+1) log k,
where the log is taken to the base 2.https://authors.library.caltech.edu/records/m4gwj-46284Ramsey numbers of sparse hypergraphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001597
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2009
DOI: 10.1002/rsa.20260
We give a short proof that any k‐uniform hypergraph H on n vertices with bounded degree Δ has Ramsey number at most c(Δ,k)n, for an appropriate constant c(Δ,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‐contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and Rödl. Moreover, our method demonstrates that, for k ≥ 4,
[equation; see abstract in PDF for details],
where the tower is of height k and the constant c depends on k. It significantly improves on the Ackermann‐type 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.https://authors.library.caltech.edu/records/4mrey-82r45A new upper bound for diagonal Ramsey numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957624
Authors: Conlon, David
Year: 2009
DOI: 10.4007/annals.2009.170.941
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant C such that
r(k + 1, k + 1) ≤ k^[-C log k/(log log k (2k_k).https://authors.library.caltech.edu/records/vjwec-s3c53Hypergraph Packing and Sparse Bipartite Ramsey Numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959718
Authors: Conlon, David
Year: 2009
DOI: 10.1017/s0963548309990174
We prove that there exists a constant c such that, for any integer Δ, the Ramsey number of a bipartite graph on n vertices with maximum degree Δ is less than 2^(cΔ)n. A probabilistic argument due to Graham, Rödl and Ruciński 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 Rödl, Ruciński and Taraz.https://authors.library.caltech.edu/records/41g6c-zjc92On-line Ramsey Numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958058
Authors: Conlon, David
Year: 2009
DOI: 10.1137/090749220
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 ˜r(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).https://authors.library.caltech.edu/records/fff0b-1g810Hypergraph Ramsey numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000262
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2010
DOI: 10.1090/s0894-0347-09-00645-6
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 ≥ 3 and s fixed. In particular, we show that
r_(3)(s, n) ≤ 2^(n^(s-2)log n),
which improves by a factor of n^(s-2)/polylog n the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant c > 0 such that
r_(3)(s, n) ≥ 2^(csn log((n/s)+1))
for all 4 ≤ s ≤ n. For constant s, this gives the first superexponential lower bound for r_(3)(s, n), answering an open question posed by Erdős 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 Erdős and Hajnal, we show that
r_(3)(n, n, n) ≥ 2^(n^(c log n)).
Finally, we make some progress on related hypergraph Ramsey-type problems.https://authors.library.caltech.edu/records/tmnj9-7nm31An Extremal Theorem in the Hypercube
https://resolver.caltech.edu/CaltechAUTHORS:20190819-163059223
Authors: Conlon, David
Year: 2010
DOI: 10.48550/arXiv.1005.0582
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 ≥ 4 other than 5.https://authors.library.caltech.edu/records/pw80h-jdy02An Approximate Version of Sidorenko's Conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958544
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2010
DOI: 10.1007/s00039-010-0097-0
A beautiful conjecture of Erdős-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.https://authors.library.caltech.edu/records/z6qfe-0dd58Large almost monochromatic subsets in hypergraphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958677
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2011
DOI: 10.1007/s11856-011-0016-6
We show that for all ℓ and ε > 0 there is a constant c = c(ℓ, ε) > 0 such that every ℓ-coloring of the triples of an N-element set contains a subset S of size c√(log N) such that at least 1 − ε 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 Erdős and Hajnal from 1989 on discrepancy in hypergraphs. For ℓ ≥ 4 colors, it is known that there is an ℓ-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only Θ(log log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an ℓ-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 ℓ-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erdős and Hajnal.https://authors.library.caltech.edu/records/jmby0-yas06Inevitable randomness in discrete mathematics [Book Review]
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958247
Authors: Conlon, David
Year: 2011
DOI: 10.1112/blms/bdr063
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.
In this book, József 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.
The central focus of the book is, to use the author's own term, the following 'vague' metaconjecture.https://authors.library.caltech.edu/records/rgb7g-cgs47Weak quasi-randomness for uniform hypergraphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001217
Authors: Conlon, David; Hàn, Hiêp; Person, Yury; Schacht, Mathias
Year: 2012
DOI: 10.1002/rsa.20389
We study quasi‐random properties of k‐uniform 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‐Graham‐Wilson theorem for quasi‐random graphs.
Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional 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‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d.https://authors.library.caltech.edu/records/7zy71-zzw53On the Ramsey multiplicity of complete graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958767
Authors: Conlon, David
Year: 2012
DOI: 10.1007/s00493-012-2465-x
We show that, for n large, there must exist at least
(n^t)/(C^((1+o(1))t^2))
monochromatic K_(t)s in any two-colouring of the edges of K_n, where C ≈ 2.18 is an explicitly defined constant. The old lower bound, due to Erdős [2], and based upon the standard bounds for Ramsey's theorem, is
(n^t)/(4^((1+o(1))t^2)).https://authors.library.caltech.edu/records/60yc2-spk66On two problems in graph Ramsey theory
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958859
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2012
DOI: 10.1007/s00493-012-2710-3
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.
The 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 Chvátal, Rödl, Szemerédi and Trotter states that there exists a constant c(Δ) such that r(H) ≤ c(Δ)n for every graph H with n vertices and maximum degree Δ. The important open question is to determine the constant c(Δ). The best results, both due to Graham, Rödl and Ruciński, state that there are positive constants c and c' such that 2^(c'Δ) ≤ c(Δ) ≤ c^(Δlog^(2)Δ). We improve this upper bound, showing that there is a constant c for which c(Δ) ≤ 2^(cΔlogΔ).https://authors.library.caltech.edu/records/s358b-c8d04Erdős-Hajnal-type theorems in hypergraphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957444
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2012
DOI: 10.1016/j.jctb.2012.05.005
The Erdős–Hajnal 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^δ(H), where δ(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.
We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is ℋ-free, for any given ℋ, then it must contain a complete or empty tripartite subgraph with parts of order c(log n)^½ + δ(ℋ), where δ(ℋ) > 0 depends only on ℋ. This improves on the bound of c(log n)^½, which holds in all 3-uniform hypergraphs, and, up to the value of the constant δ(ℋ), is best possible.
We also prove that, for k ≥ 4, no analogue of the standard Erdős–Hajnal conjecture can hold in k-uniform hypergraphs. That is, there are k-uniform hypergraphs ℋ and sequences of ℋ-free hypergraphs which do not contain cliques or independent sets of size appreciably larger than one would normally expect.https://authors.library.caltech.edu/records/69ssa-jy057Bounds for graph regularity and removal lemmas
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959355
Authors: Conlon, David; Fox, Jacob
Year: 2012
DOI: 10.1007/s00039-012-0171-x
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 ϵ-regular, where c, ϵ > 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 Szemerédi'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 ϵ may require as many as 2^Ω(ϵ^(−2)) parts. This is tight up to the implied constant and solves a problem studied by Lovász and Szegedy.https://authors.library.caltech.edu/records/5dhpf-yek21The Ramsey number of dense graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170836067
Authors: Conlon, David
Year: 2013
DOI: 10.1112/blms/bds097
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 ρ, proving that r(H) ≤ 2^(c√(ρ)log(2/ρ)t). We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree ρt and the Ramsey number of random graphs in G(t, ρ), that is, graphs on t vertices where each edge has been chosen independently with probability ρ.https://authors.library.caltech.edu/records/sp7g8-vxr43An improved bound for the stepping-up lemma
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957165
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2013
DOI: 10.1016/j.dam.2010.10.013
The partition relation N → (n)^(k)_(ℓ) means that whenever the k-tuples of an N-element set are ℓ-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 → (n)^(k)_(ℓ) is written as N / → (n)^(k)_(ℓ). An ingenious construction of Erdős 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 ℓ ≥ 2, k ≥ 3, and N / → (n)^(k)_(ℓ), then 2^N / → (2n + k - 4)^(k+1)_(ℓ). In this paper we give an improved construction for k ≥ 4. We introduce a general class of colorings which extends the framework of Erdős and Hajnal and can be used to establish negative partition relations. We show that if ℓ ≥ 2, k ≥ 4 and N / → (n)^(k)_(ℓ), then 2^N / → (n + 3)^(k+1)_(ℓ). If also k is odd or ℓ ≥ 3, then we get the better bound 2^N / → (n + 2)^(k+1)_(ℓ). This improved bound gives a coloring of the k-tuples whose largest monochromatic set is a factor Ω(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 ℓ ≥ 4 we determine up to an absolute constant factor (which is independent of k) the size of the largest guaranteed monochromatic set in an ℓ-coloring of the k-tuples of an N-set.https://authors.library.caltech.edu/records/5hz8p-e8k15Two extensions of Ramsey's theorem
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959551
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2013
DOI: 10.1215/00127094-2382566
Ramsey's theorem, in the version of Erdős 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 Rödl, 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 ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds: for every permutation π of 1, . . ., k−1, every 2-coloring of the edges of the complete graph on {1, 2, . . ., n} contains a monochromatic clique a_1 < • • • < a_k with
a_(π(1)+1) − a_(π(1)) > a_π((2)+1) − a_(π(2)) > • • • > a_(π(k−1)+1) − a_(π(k−1)).
That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, 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.https://authors.library.caltech.edu/records/ryxaq-f1122The Green-Tao theorem: an exposition
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001431
Authors: Conlon, David; Fox, Jacob; Zhao, Yufei
Year: 2014
DOI: 10.4171/emss/6
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.https://authors.library.caltech.edu/records/8zjkv-zvw34Short Proofs of Some Extremal Results
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162956937
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2014
DOI: 10.1017/s0963548313000448
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.https://authors.library.caltech.edu/records/ew3yz-n7p38Extremal results in sparse pseudorandom graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957263
Authors: Conlon, David; Fox, Jacob; Zhao, Yufei
Year: 2014
DOI: 10.1016/j.aim.2013.12.004
Szemerédi'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 Rödl proved an analogue of Szemerédi'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 Szemerédi'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.
The 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 Rödl, 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 Erdős–Stone–Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.https://authors.library.caltech.edu/records/2a4a3-ndy66Ramsey-type results for semi-algebraic relations
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000075
Authors: Conlon, David; Fox, Jacob; Pach, János; Sudakov, Benny; Suk, Andrew
Year: 2014
DOI: 10.1090/s0002-9947-2014-06179-5
A k-ary semi-algebraic relation E on ℝ_d is a subset of ℝ_(kd), the set of k-tuples of points in ℝ_(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 ≤ t and the number of polynomials and their degrees are all bounded by t. A set A ⊂ ℝ_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 ℝ_d equipped with semi-algebraic relations.
In 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 − 1. This improves upon a direct application of Ramsey's theorem by one exponential and extends a result of Alon, Pach, Pinchasi, Radoičić, 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.https://authors.library.caltech.edu/records/0hxg6-rgb55On the KŁR conjecture in random graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958948
Authors: Conlon, D.; Gowers, W. T.; Samotij, W.; Schacht, M.
Year: 2014
DOI: 10.1007/s11856-014-1120-1
The KŁR conjecture of Kohayakawa, Łuczak, and Rödl 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 Rödl. 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.https://authors.library.caltech.edu/records/pfh4w-3qv46Cycle packing
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001129
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2014
DOI: 10.1002/rsa.20574
In the 1960s, Erdős 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 Erdős‐Gallai conjecture for random graphs and for graphs with linear minimum degree.https://authors.library.caltech.edu/records/2xrg4-tje16The Erdős-Gyárfás problem on generalized Ramsey numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959803
Authors: Conlon, David; Fox, Jacob; Lee, Choongbum; Sudakov, Benny
Year: 2015
DOI: 10.1112/plms/pdu049
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 Erdős and Gyárfás 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.https://authors.library.caltech.edu/records/mrfhv-pps81Distinct Volume Subsets
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958151
Authors: Conlon, David; Fox, Jacob; Gasarch, William; Harris, David G.; Ulrich, Douglas; Zbarsky, Samuel
Year: 2015
DOI: 10.1137/140954519
Suppose that a and d are positive integers with a ≥ 2. Let h_(a,d)(n) be the largest integer t such that any set of n points in ℝ^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 Erdős 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.https://authors.library.caltech.edu/records/p1qtp-rka12A relative Szemerédi theorem
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959067
Authors: Conlon, David; Fox, Jacob; Zhao, Yufei
Year: 2015
DOI: 10.1007/s00039-015-0324-9
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 Szemerédi 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 Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for k-term arithmetic progressions in pseudorandom subsets of ℤ_N of density N^(−ck). 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 Szemerédi theorem.https://authors.library.caltech.edu/records/fecwr-jw093On the Grid Ramsey Problem and Related Questions
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959982
Authors: Conlon, David; Fox, Jacob; Lee, Choongbum; Sudakov, Benny
Year: 2015
DOI: 10.1093/imrn/rnu190
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.https://authors.library.caltech.edu/records/mrmac-73f25Almost-spanning universality in random graphs (Extended abstract)
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957531
Authors: Conlon, David; Ferber, Asaf; Nenadov, Rajko; Škorić, Nemanja
Year: 2015
DOI: 10.1016/j.endm.2015.06.030
A graph G is said to be ℋ(n, Δ)-universal if it contains every graph on n vertices with maximum degree at most Δ. It is known that for any ε > 0 and any natural number Δ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely ℋ((1 - ε)n, Δ)-universal for p ≥ c(log n/n)^(1/Δ). Bypassing this natural boundary Δ ≥ 3, we show that for the same conclusion holds when [equation; see abstract in PDF for details].https://authors.library.caltech.edu/records/r5egw-tmm70Ramsey numbers of cubes versus cliques
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959158
Authors: Conlon, David; Fox, Jacob; Lee, Choongbum; Sudakov, Benny
Year: 2016
DOI: 10.1007/s00493-014-3010-x
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 Erdős asked whether the simple lower bound r(Q_n, K_s) ≥ (s−1)(2^(n) − 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.https://authors.library.caltech.edu/records/q1vhm-qth77Monochromatic Cycle Partitions in Local Edge Colorings
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163001038
Authors: Conlon, David; Stein, Maya
Year: 2016
DOI: 10.1002/jgt.21867
An edge coloring of a graph is said to be an r‐local coloring if the edges incident to any vertex are colored with at most r colors. Generalizing a result of Bessy and Thomassé, we prove that the vertex set of any 2‐locally 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‐locally colored complete graph may be partitioned into O(r^(2) log r) disjoint monochromatic cycles. This generalizes a result of Erdős, Gyárfás, and Pyber.https://authors.library.caltech.edu/records/985j8-0fe05Combinatorial theorems in sparse random sets
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957976
Authors: Conlon, D.; Gowers, W. T.
Year: 2016
DOI: 10.4007/annals.2016.184.2.2
We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turán's theorem, Szemerédi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Turán's theorem to the random setting by showing that for every ϵ > 0 and every positive integer t ≥ 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^(−2/(t+1)), then, with probability tending to 1 as n tends to infinity, every subgraph of G with at least (1 – (1/(t−1)) + ϵ)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 Turán 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, Rödl and Schacht.https://authors.library.caltech.edu/records/ahtz2-11b03Short proofs of some extremal results II
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000938
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2016
DOI: 10.1016/j.jctb.2016.03.005
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.https://authors.library.caltech.edu/records/csf5h-1bf02Ordered Ramsey numbers
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000833
Authors: Conlon, David; Fox, Jacob; Lee, Choongbum; Sudakov, Benny
Year: 2017
DOI: 10.1016/j.jctb.2016.06.007
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) ≤ r(H)^(c log^(2) n) for any labeled graph H on vertex set {1, 2 . . ., n}.https://authors.library.caltech.edu/records/v2sqk-x4v76Freiman homomorphisms on sparse random sets
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000651
Authors: Conlon, D.; Gowers, W. T.
Year: 2017
DOI: 10.1093/qmath/haw058
A result of Fiz Pontiveros shows that if A is a random subset of ℤ_N where each element is chosen independently with probability N^(−1/2+o(1)), then with high probability every Freiman homomorphism defined on A can be extended to a Freiman homomorphism on the whole of ℤ_N. In this paper, we improve the bound to CN^(−2/3)(logN)^(1/3), which is best possible up to the constant factor.https://authors.library.caltech.edu/records/rqpp7-81c15A Sequence of Triangle-Free Pseudorandom Graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000736
Authors: Conlon, David
Year: 2017
DOI: 10.1017/s0963548316000298
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.https://authors.library.caltech.edu/records/g4na6-qn093Quasirandom Cayley graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170904052
Authors: Conlon, David; Zhao, Yufei
Year: 2017
DOI: 10.19086/da.1294
We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, Rödl, 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.https://authors.library.caltech.edu/records/t4b2v-c5r37Almost-spanning universality in random graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000547
Authors: Conlon, David; Ferber, Asaf; Nenadov, Rajko; Škorić, Nemanja
Year: 2017
DOI: 10.1002/rsa.20661
A graph G is said to be ℋ(n, Δ)-universal if it contains every graph on n vertices with maximum degree at most Δ. It is known that for any ε > 0 and any natural number Δ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely ℋ((1 - ε)n, Δ)-universal for p ≥ c(log n/n)^(1/Δ). Bypassing this natural boundary Δ ≥ 3, we show that for the same conclusion holds when [equation; see abstract in PDF for details].https://authors.library.caltech.edu/records/fd3nb-r4991Finite reflection groups and graph norms
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162957357
Authors: Conlon, David; Lee, Joonkyung
Year: 2017
DOI: 10.1016/j.aim.2017.05.009
Given a graph H on vertex set {1, 2, • • •, n} and a function f : [0, 1]^2 → ℝ, define [equation; see abstract in PDF for details], where μ is the Lebesgue measure on [0, 1]. We say that H is norming if ∥•∥_H is a semi-norm. A similar notion ∥•∥_r(H) is defined by ∥f∥_r(H) := ∥|f|∥_H and H is said to be weakly norming if ∥•∥_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.https://authors.library.caltech.edu/records/85q64-q6758Quasirandomness in hypergraphs
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000449
Authors: Aigner-Horev, Elad; Conlon, David; Hàn, Hiệp; Person, Yury; Schacht, Mathias
Year: 2017
DOI: 10.1016/j.endm.2017.06.015
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.
In 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.https://authors.library.caltech.edu/records/ehfed-mgm13Rational exponents in extremal graph theory
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000355
Authors: Bukh, Boris; Conlon, David
Year: 2018
DOI: 10.4171/jems/798
Given a family of graphs ℌ, the extremal number ex(n, ℌ) is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family ℌ as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs ℌ_r such that ex (n, ℌ_r) = Θ(n^r). This solves a longstanding problem in the area of extremal graph theory.https://authors.library.caltech.edu/records/fjteg-4ys50Hereditary quasirandomness without regularity
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959631
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2018
DOI: 10.1017/s0305004116001055
A result of Simonovits and Sós states that for any fixed graph H and any ϵ > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains p^(e(H))|S|^(v(H)) ± δn^(v(H)) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains ½p|S|^2 ± ϵn^2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ^(−1) which is a tower of twos of height polynomial in ϵ^(−1). We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ϵ when H is a clique and polynomial in ϵ for general H. This answers a problem raised by Simonovits and Sós.https://authors.library.caltech.edu/records/xr8k2-9be11Quasirandomness in Hypergraphs
https://resolver.caltech.edu/CaltechAUTHORS:20190821-110928639
Authors: Aigner-Horev, Elad; Conlon, David; Hàn, Hiệp; Person, Yury; Schacht, Mathias
Year: 2018
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.
In 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.https://authors.library.caltech.edu/records/vpw4b-fp197Some advances on Sidorenko's conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958339
Authors: Conlon, David; Kim, Jeong Han; Lee, Choongbum; Lee, Joonkyung
Year: 2018
DOI: 10.1112/jlms.12142
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.https://authors.library.caltech.edu/records/ysvq9-q6365Lines in Euclidean Ramsey Theory
https://resolver.caltech.edu/CaltechAUTHORS:20190812-162959449
Authors: Conlon, David; Fox, Jacob
Year: 2019
DOI: 10.1007/s00454-018-9980-5
Let ℓ_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 ℓ_2 and no blue copy of ℓ_m for any m ≥ 2^(cn). This is best possible up to the constant c in the exponent. It also answers a question of Erdős et al. (J Comb Theory Ser A 14:341–363, 1973). They asked if, for every natural number n, there is a set K ⊂ E^1 and a red/blue-coloring of E^n containing no red copy of ℓ_2 and no blue copy of K.https://authors.library.caltech.edu/records/qqfbd-zpw07Tower-type bounds for unavoidable patterns in words
https://resolver.caltech.edu/CaltechAUTHORS:20190812-163000172
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2019
DOI: 10.1090/tran/7751
A word ω 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 ω. 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 − 1x_(n)Z_(n) − 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) = Θ(2_(q)q!).https://authors.library.caltech.edu/records/ctm7x-9at88Books versus triangles at the extremal density
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170946478
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2020
DOI: 10.1137/19M1261766
A celebrated result of Mantel shows that every graph on n vertices with [n²/4] + 1 edges must contain a triangle. A robust version of this result, due to Rademacher, says that there
must, in fact, be at least [n/2] triangles in any such graph. Another strengthening, due to the
combined efforts of many authors starting with Erdős, says that any such graph must have an edge
which is contained in at least n/6 triangles. Following Mubayi, we study the interplay between
these two results, that is, between the number of triangles in such graphs and their book number,
the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any
1/6 ≤ β < 1/4 there is γ > 0 such that any graph on n vertices with at least [n²/4] +1 edges and book number at most βn contains at least (γ - o(1))n³ triangles. He also asked for a more precise
estimate for γ in terms of β. We make a conjecture about this dependency and prove this conjecture
for β = 1/6 and for 0.2495 ≤ β < 1/4, thereby answering Mubayi's question in these ranges.https://authors.library.caltech.edu/records/n7nss-psj53Ramsey Numbers of Books and Quasirandomness
https://resolver.caltech.edu/CaltechAUTHORS:20200914-140122041
Authors: Conlon, David; Fox, Jacob; Wigderson, Yuval
Year: 2020
DOI: 10.1007/s00493-021-4409-9
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 Erdős, 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 Szemerédi'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.https://authors.library.caltech.edu/records/pqpqt-9ba35Hypergraph expanders of all uniformities from Cayley graphs
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170932806
Authors: Conlon, David; Tidor, Jonathan; Zhao, Yufei
Year: 2020
DOI: 10.1112/plms.12371
Hypergraph expanders are hypergraphs with surprising, non‐intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of 3‐uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of r‐uniform hypergraph expanders for all r ⩾ 3.https://authors.library.caltech.edu/records/hbkkk-d8404Short proofs of some extremal results III
https://resolver.caltech.edu/CaltechAUTHORS:20200909-070916091
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny
Year: 2020
DOI: 10.1002/rsa.20953
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.https://authors.library.caltech.edu/records/scafr-q3881Ramsey games near the critical threshold
https://resolver.caltech.edu/CaltechAUTHORS:20190820-162204728
Authors: Conlon, David; Das, Shagnik; Lee, Joonkyung; Mészáros, Tamás
Year: 2020
DOI: 10.1002/rsa.20959
A well‐known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if p ≥ Cn^(-1/m2(H)), then the random graph G_(n, p) is a.a.s. H‐Ramsey, that is, any 2‐coloring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0‐statement also holds, that is, there exists c > 0 such that whenever p ≤ Cn^(-1/m2(H)) the random graph Gn, p is a.a.s. not H‐Ramsey. We show that near this threshold, even when G_(n, p) is not H‐Ramsey, it is often extremely close to being H‐Ramsey. More precisely, we prove that for any constant c > 0 and any strictly 2‐balanced graph H, if p ≥ Cn^(-1/m2(H)), then the random graph G_(n, p) a.a.s. has the property that every 2‐edge‐coloring without monochromatic copies of H cannot be extended to an H‐free coloring after ω(1) extra random edges are added. This generalizes a result by Friedgut, Kohayakawa, Rödl, Ruciński, 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‐color case and show that these theorems need not hold when H is not strictly 2‐balanced.https://authors.library.caltech.edu/records/f781n-d0219Lower bounds for multicolor Ramsey numbers
https://resolver.caltech.edu/CaltechAUTHORS:20201216-111908862
Authors: Conlon, David; Ferber, Asaf
Year: 2021
DOI: 10.1016/j.aim.2020.107528
We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.https://authors.library.caltech.edu/records/5k8hv-4ts49Extremal Numbers of Cycles Revisited
https://resolver.caltech.edu/CaltechAUTHORS:20210602-132638631
Authors: Conlon, David
Year: 2021
DOI: 10.1080/00029890.2021.1886845
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.https://authors.library.caltech.edu/records/8w53p-t0j17More on the extremal number of subdivisions
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170943053
Authors: Conlon, David; Janzer, Oliver; Lee, Joonkyung
Year: 2021
DOI: 10.1007/s00493-020-4202-1
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′_(s,t) for the subdivision of the bipartite graph K_(s,t), we show that ex(n,K′_(s,t)) = O(n^(3/2)−1/2s)). 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 ≥ 1, we show that ex(n,L) = Θ(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 Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(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 ≥ 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 δ > 0 such that ex(n, H^(k−1)) = O(^(n1+1/k−δ)), 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 C₄ as a subgraph satisfies ex(n, H) = o(n^(2−1/r)).https://authors.library.caltech.edu/records/sqbep-vc039Random Multilinear Maps and the Erdős Box Problem
https://resolver.caltech.edu/CaltechAUTHORS:20211105-180659290
Authors: Conlon, David; Pohoata, Cosmin; Zakharov, Dmitriy
Year: 2021
DOI: 10.19086/da.28336
By using random multilinear maps, we provide new lower bounds for the Erdős
box problem, the problem of estimating the extremal number of the complete d-partite duniform
hypergraph with two vertices in each part, thereby improving on work of Gunderson,
Rödl and Sidorenko.https://authors.library.caltech.edu/records/mzy87-2zv57Repeated Patterns in Proper Colorings
https://resolver.caltech.edu/CaltechAUTHORS:20200914-134941914
Authors: Conlon, David; Tyomkyn, Mykhaylo
Year: 2021
DOI: 10.1137/21M1414103
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 ω(1) repeats of T.https://authors.library.caltech.edu/records/ypmvs-51734The regularity method for graphs with few 4-cycles
https://resolver.caltech.edu/CaltechAUTHORS:20200914-101307280
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny; Zhao, Yufei
Year: 2021
DOI: 10.1112/jlms.12500
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:
Every n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n^(3/2)) edges.
For r ⩾ 3, every n-vertex r-graph with girth greater than 5 has o(n^(3/2)) edges.
Every subset of [n] without a nontrivial solution to the equation x₁ + x₂ + 2x₃ = x₄ + 3x₅ has size o(√n).https://authors.library.caltech.edu/records/137hj-tjx44Which graphs can be counted in C₄-free graphs?
https://resolver.caltech.edu/CaltechAUTHORS:20230515-296699000.2
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny; Zhao, Yufei
Year: 2022
DOI: 10.4310/pamq.2022.v18.n6.a4
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://authors.library.caltech.edu/records/5bfv1-vtn47Monochromatic combinatorial lines of length three
https://resolver.caltech.edu/CaltechAUTHORS:20190819-170936221
Authors: Conlon, David
Year: 2022
DOI: 10.1090/proc/15739
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.https://authors.library.caltech.edu/records/fstjd-qqb57Some remarks on the Zarankiewicz problem
https://resolver.caltech.edu/CaltechAUTHORS:20200914-085046091
Authors: Conlon, David
Year: 2022
DOI: 10.1017/S0305004121000475
The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1's in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1's. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán 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.https://authors.library.caltech.edu/records/cq695-mc662The size‐Ramsey number of cubic graphs
https://resolver.caltech.edu/CaltechAUTHORS:20220718-901273500
Authors: Conlon, David; Nenadov, Rajko; Trujić, Miloš
Year: 2022
DOI: 10.1112/blms.12682
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, Rödl, Schacht, and Szemerédi. 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⩾C_n^(−2/5) is with high probability Ramsey for any cubic graph with n vertices. This latter result is best possible up to the constant.https://authors.library.caltech.edu/records/tg42p-xc421Ramsey numbers of trails and circuits
https://resolver.caltech.edu/CaltechAUTHORS:20220726-997455000
Authors: Conlon, David; Tyomkyn, Mykhaylo
Year: 2022
DOI: 10.1002/jgt.22865
We show that every two-colouring of the edges of the complete graph Kₙ contains a monochromatic trail or circuit of length at least 2n²/9+o(n²), which is asymptotically best possible.https://authors.library.caltech.edu/records/63bb7-54c82Threshold Ramsey multiplicity for odd cycles
https://resolver.caltech.edu/CaltechAUTHORS:20221107-997760900.3
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny; Wei, Fan
Year: 2022
DOI: 10.33044/revuma.2874
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 Kₙ 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)ᵏ, 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.https://authors.library.caltech.edu/records/kd92j-5zw50The upper logarithmic density of monochromatic subset sums
https://resolver.caltech.edu/CaltechAUTHORS:20221017-10817000.4
Authors: Conlon, David; Fox, Jacob; Pham, Huy Tuan
Year: 2022
DOI: 10.1112/mtk.12167
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 + √3)/4 and this is best possible. This answers a 40-year-old question of Erdős.https://authors.library.caltech.edu/records/3bzbc-xj302Threshold Ramsey multiplicity for paths and even cycles
https://resolver.caltech.edu/CaltechAUTHORS:20221011-128968500.8
Authors: Conlon, David; Fox, Jacob; Sudakov, Benny; Wei, Fan
Year: 2023
DOI: 10.1016/j.ejc.2022.103612
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 Kₙ 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)ᵏ. This bound is tight up to the constant c. We prove a similar result for odd cycles in a companion paper.https://authors.library.caltech.edu/records/v1t30-85j35A new bound for the Brown-Erdős-Sós problem
https://resolver.caltech.edu/CaltechAUTHORS:20221031-575177800.7
Authors: Conlon, David; Gishboliner, Lior; Levanzov, Yevgeny; Shapira, Asaf
Year: 2023
DOI: 10.1016/j.jctb.2022.08.005
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) such 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(log e / log log e), e) = o(n²).https://authors.library.caltech.edu/records/hjwhg-sd610On the size-Ramsey number of grids
https://resolver.caltech.edu/CaltechAUTHORS:20230717-55915200.33
Authors: Conlon, David; Nenadov, Rajko; Trujić, Miloš
Year: 2023
DOI: 10.1017/s0963548323000147
We show that the size-Ramsey number of the √n × √n grid graph is O(n^(5/4)), improving a previous bound of n^(3/2 + o(1)) by Clemens, Miralaei, Reding, Schacht, and Taraz.https://authors.library.caltech.edu/records/z589a-6dp27Sums of transcendental dilates
https://resolver.caltech.edu/CaltechAUTHORS:20230725-500420000.2
Authors: Conlon, David; Lim, Jeck
Year: 2023
DOI: 10.1112/blms.12870
We show that there is an absolute constant c > 0 such that |A + λ • A| ⩾ e^(c√log|A)| |A| for any finite subset A of ℝ and any transcendental number λ ∈ ℝ. By a construction of Konyagin and Łaba, this is best possible up to the constant c.https://authors.library.caltech.edu/records/ynbpx-c4633