Threshold Functions for Ramsey Properties

Rödl, Vojtěch; Ruciński, Andrzej

doi:10.2307/2152833

Cited by 140 publications

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“…By transferring the problem into a random setting, Rödl and Ruciński [18] showed that in fact such graphs G are quite frequent. More precisely, they proved the following result.…”

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confidence: 99%

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Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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Consider the following problem: For given graphs G and F1,…,Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Ruciński studied this problem for the random graph Gn,p in the symmetric case when k is fixed and F1 = ··· = Fk = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p ≤ bn−β for some constants b = b(F,k) and β = β(F). This result is essentially best possible because for p ≥ Bn−β, where B = B(F,k) is a large constant, such an edge‐coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n for arbitrary F1,…,Fk.In this article we address the case when F1,…,Fk are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k‐edge‐coloring of Gn,p with p ≤ bn−β for some constant b = b(F1,…,Fk), where β = β(F1,…,Fk) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F1,…,Fk) such that for p ≥ Bn−β the random graph Gn,p a.a.s. does not have a valid k‐edge‐coloring provided the so‐called KŁR‐conjecture holds. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

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“…By transferring the problem into a random setting, Rödl and Ruciński [18] showed that in fact such graphs G are quite frequent. More precisely, they proved the following result.…”

mentioning

confidence: 99%

“…Recall that in the binomial random graph G n,p on n vertices, every edge is present with probability 0 ≤ p = p(n) ≤ 1 independently of all other edges. Theorem 1 ( [13], [17], [18]). Let k ≥ 2 and F be a non-empty graph that is not a forest.…”

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confidence: 99%

Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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“…It is well known that sufficiently large ( , d)-dense graphs contain any fixed subgraph (see eg, [19]). The Sparse Counting Lemma (Lemma 22) applied with F and d CL = λ/4 yields constants ε CL > 0 and ξ CL > 0.…”

Section: Proof Of Lemma 23mentioning

confidence: 99%

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

Schacht

Schulenburg

2017

Random Struct Algorithms

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“…Some of the early highlights include a version of Mantel's theorem for random graphs proved by Babai, Simonovits and Spencer [1], the Ramsey theoretic results of Rödl and Ruciński [25,26], and a random analogue of Szemerédi's theorem due to Kohayakawa, Łuczak and Rödl [20]. Very general transference theorems have since been proved by Conlon and Gowers [8], Schacht [29], Balogh, Morris and Samotij [4], and Saxton and Thomason [28].…”

Section: Introductionmentioning

confidence: 99%

Transference for the Erdős–ko–rado Theorem

2015

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For natural numbers n, r ∈ N with n r , the Kneser graph K (n, r ) is the graph on the family of r -element subsets of {1, . . . , n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K (n, r ) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as r/n is bounded away from 1/2, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős-Ko-Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

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Threshold Functions for Ramsey Properties

Cited by 140 publications

References 0 publications

Asymmetric Ramsey properties of random graphs involving cliques

Asymmetric Ramsey properties of random graphs involving cliques

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

Transference for the Erdős–ko–rado Theorem

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