A Short Proof of the Random Ramsey Theorem

Nenadov, Rajko; Steger, Angelika

doi:10.1017/s0963548314000832

Cited by 50 publications

(95 citation statements)

References 10 publications

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“…We do this in the remainder of this section. Actually, our proof of Lemma follows the proof of Lemma 3.1 from . The main difference is that in a problematic copy of H was defined as a copy of H in which all edges are contained in two copies of H , while the definition in this paper allows the existence of one (but only one) edge that may be open.…”

Section: Proof Of the 0‐statement Of Theoremmentioning

confidence: 99%

“…Actually, our proof of Lemma follows the proof of Lemma 3.1 from . The main difference is that in a problematic copy of H was defined as a copy of H in which all edges are contained in two copies of H , while the definition in this paper allows the existence of one (but only one) edge that may be open. As we shall see, this difference in definition is responsible for the fact that the proof goes through for triangles in , but does not here.…”

Section: Proof Of the 0‐statement Of Theoremmentioning

confidence: 99%

“…The main difference is that in a problematic copy of H was defined as a copy of H in which all edges are contained in two copies of H , while the definition in this paper allows the existence of one (but only one) edge that may be open. As we shall see, this difference in definition is responsible for the fact that the proof goes through for triangles in , but does not here. Of course, this is no coincidence: for the Random Ramsey result that was considered in , the threshold for triangles is of order

n^{- 1 / m_{2} (K_{3})} = n^{- 1 / 2}

(Theorem ), while for the Maker‐Breaker game considered in this paper the threshold for triangles is

n^{- 5 / 9}

(Theorem ).…”

Section: Proof Of the 0‐statement Of Theoremmentioning

confidence: 99%

“…Of all the known results guaranteeing Maker's win in the H ‐game (played on various base graphs), this is the first case of an explicit winning strategy. The proof of the 0‐statement is based on ideas from .…”

Section: Introductionmentioning

confidence: 99%

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On the threshold for the Maker‐Breaker H‐game

Nenadov

Steger

Stojaković

2015

Random Struct Algorithms

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Abstract. We study the Maker-Breaker H-game played on the edge set of the random graph Gn,p. In this game two players, Maker and Breaker, alternately claim unclaimed edges of Gn,p, until all the edges are claimed. Maker wins if he claims all the edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H-game is given by the threshold of the corresponding Ramsey property of Gn,p with respect to the graph H.

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Section: Proof Of the 0‐statement Of Theoremmentioning

confidence: 99%

Section: Proof Of the 0‐statement Of Theoremmentioning

confidence: 99%

n^{- 1 / m_{2} (K_{3})} = n^{- 1 / 2}

(Theorem ), while for the Maker‐Breaker game considered in this paper the threshold for triangles is

n^{- 5 / 9}

(Theorem ).…”

Section: Proof Of the 0‐statement Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the threshold for the Maker‐Breaker H‐game

Nenadov

Steger

Stojaković

2015

Random Struct Algorithms

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“…Our proof is a generalization of an approach from [11] to hypergraphs and general Ramsey problems. It is essentially a first moment argument.…”

Section: Proof Of Theorem 18mentioning

confidence: 95%

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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Chromatic number is Ramsey distinguishing

Savery

2021

Journal of Graph Theory

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A graph G is Ramsey for a graph H if every colouring of the edges of G in two colours contains a monochromatic copy of H. Two graphs H 1 and H 2 are Ramsey equivalent if any graph G is Ramsey for H 1 if and only if it is Ramsey for H 2. A graph parameter s is Ramsey distinguishing if s ( H 1 ) ≠ s ( H 2 ) implies that H 1 and H 2 are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multicolour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.

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A Short Proof of the Random Ramsey Theorem

Cited by 50 publications

References 10 publications

On the threshold for the Maker‐Breaker H‐game

On the threshold for the Maker‐Breaker H‐game

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

Chromatic number is Ramsey distinguishing

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