Ramsey properties of random discrete structures

Friedgut, Ehud; Rödl, Vojtěch; Schacht, Mathias

doi:10.1002/rsa.20352

Cited by 49 publications

(101 citation statements)

References 33 publications

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“…Then, the right order of magnitude for the lower bound for t is n 1−1/mA . Let us also mention that this parameter m A was already exploited in the work of Schacht [33], Friedgut, Rödl and Schacht [13], and Saxton and Thomason [31]. We now show that our specification of t in terms of the different coefficients {α We can study then the modified parameter (27) max…”

Section: 3mentioning

confidence: 65%

Counting configuration–free sets in groups

Rué

Serra

Vena

2015

Electronic Notes in Discrete Mathematics

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Abstract. We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. As a consequence, random counterparts are presented as well.

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Section: 3mentioning

confidence: 65%

Counting configuration–free sets in groups

Rué

Serra

Vena

2015

Electronic Notes in Discrete Mathematics

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“…Here we determine the threshold for all complete -uniform hypergraphs of size at least + 2. As lower bounds for the 2-bounded anti-Ramsey problem imply lower bounds for the classical Ramsey problem, this result in turn also provides a matching lower bound for the upper bounds from [5,3] in the case of -uniform hypergraph cliques of size at least + 2. The remaining case ofuniform hypergraph cliques of size +1 we prove directly using the framework, thus completely characterizing edge probabilities for the Ramsey problem in the case of hypergraph cliques.…”

Section: Introductionmentioning

confidence: 80%

“…However, the corresponding problem for hypergraphs remained open for more than 15 years. Only recently, Friedgut, Rödl and Schacht [5] and independently Conlon and Gowers [3] obtained an upper bound which is an analogue to the graph case. However, the question whether there exists a matching lower bound remained open.…”

Section: Introductionmentioning

confidence: 99%

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

Nenadov¹,

Škorić²,

Steger³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that finds the desired colouring with high probability. Our framework allows to reduce the probabilistic problem of whether the Ramsey property at hand holds for random (hyper)graphs with edge probability p to a deterministic question of whether there exists a finite graph that forms an obstruction. In the second part of the paper we apply this framework to address and solve various open problems. In particular, we provide a matching lower bound for the result of Friedgut, Rödl and Schacht (2010) and, independently, Conlon and Gowers (2014+) for the classical Ramsey problem for hypergraphs in the case of cliques. A problem that was open for more than 15 years. We also improve a result of Bohman, Frieze, Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs and extend it to hypergraphs. Finally, we provide matching lower bounds for a proper-colouring version of anti-Ramsey problems introduced by Kohayakawa, .

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“…Let us remark that the methods from [22,102] also can be used to derive thresholds for Ramsey properties for random hypergraphs and random subsets of the integers (see [22,45] for details).…”

Section: Discussionmentioning

confidence: 99%

Extremal Results in Random Graphs

Rödl

Schacht

2013

Bolyai Society Mathematical Studies

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Abstract. According to Paul Erdős [Some notes on Turán's mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Turán who "created the area of extremal problems in graph theory". However, without a doubt, Paul Erdős popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and all over the world. In fact, most of the early contributions in this field can be traced back to Paul Erdős, Paul Turán, as well as their collaborators and students. Paul Erdős also established the probabilistic method in discrete mathematics, and in collaboration with Alfréd Rényi, he started the systematic study of random graphs. We shall survey recent developments at the interface of extremal combinatorics and random graph theory.

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Ramsey properties of random discrete structures

Cited by 49 publications

References 33 publications

Counting configuration–free sets in groups

Counting configuration–free sets in groups

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

Extremal Results in Random Graphs

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