An improved lower bound on multicolor Ramsey numbers

Wigderson, Yuval

doi:10.1090/proc/15447

Cited by 15 publications

(16 citation statements)

References 10 publications

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“…Our first lemma relates P(t, t) to the multicolor Ramsey numbers. It is a slight variant of [10,Lemma 3.1].…”

Section: Lower Bounds and Application To Multicolor Ramsey Numbersmentioning

confidence: 99%

An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

Sawin¹

2021

Preprint

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The multicolor Ramsey number problem asks, for each pair of natural numbers ℓ and t, for the largest ℓ-coloring of a complete graph with no monochromatic clique of size t. Recent work of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. In addition, we formalize the problem of producing a graph useful for this argument as the half-multiplicity Ramsey problem, and prove basic lower and upper bounds.Let r(t; ℓ) be the multicolor Ramsey number, that is, the minimum r such that every ℓ-coloring of the edges of the complete graph K r on r vertices has a monochromatic clique of size t.Building on a breakthrough of Conlon and Ferber [4], Wigderson [9] proved the lower bound r(t; ℓ) ≥ 2 3(ℓ−2)t 8+ t 2 +o(t) .

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“…Our first lemma relates P(t, t) to the multicolor Ramsey numbers. It is a slight variant of [10,Lemma 3.1].…”

Section: Lower Bounds and Application To Multicolor Ramsey Numbersmentioning

confidence: 99%

An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

Sawin¹

2021

Preprint

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“…In [20], Wigderson pointed out that the random induced subgraphs and random blowups are closely related and are both part of a more general framework of random homomorphisms. We do agree that the framework of random homomorphisms is great, and in some sense we can regard the random induced subgraphs and the random blowups as the same thing.…”

Section: Remarksmentioning

confidence: 99%

“…However, in this paper, we prefer to use the random blowups method, since our C 5 -free base graph from the random block construction has too many independent sets of small order. In diagonal Ramsey case [6,20], the base graph is constructed algebraically, which provides a good estimation of the number of small independent sets. The random blowups method has certain limitations, for example, it cannot work when the graph H is bipartite, unless one chooses other ways to define the "blowups".…”

Section: Remarksmentioning

confidence: 99%

“…For more literature on the Ramsey problems, we refer the readers to the dynamic survey [14] and the references therein. Inspired by the random homomorphisms method in the recent work [20] on multicolor diagonal Ramsey number, we prove the lower bound for R k (H; K m ) when H is C 5 or C 7 , respectively. The appropriate base graphs come from the random block construction in [8].…”

Section: Introductionmentioning

confidence: 99%

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A note on multicolor Ramsey number of small odd cycles versus a large clique

Xu¹,

Ge²

2021

Preprint

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Let R k (H; K m ) be the smallest number N such that every coloring of the edges of K N with k + 1 colors has either a monochromatic H in color i for some 1 i k, or a monochromatic K m in color k + 1. In this short note, we study the lower bound for3k 8 +1 ), and R k (C 7 ; K m ) = Ω(m 2k 9 +1 /(log m) 2k 9 +1 ), for fixed positive integer k and m → ∞. These slightly improve the previously known lower bound R k (C 2ℓ+1 ; K m ) = Ω(m k 2ℓ−1 +1 /(log m) k+ 2k 2ℓ−1 ) obtained by Alon and Rödl. The proof is based on random block constructions and random blowups argument.

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“…Recently, Conlon and Ferber [46] proved a lower bound r k (t) ≥ 2 7k/24+C t−o (t) , where C depends on the residue of k modulo 3. Soon afterwards, this lower bound was improved to r k (t) ≥ 2 3k/8−1/4 t−o (t) by Wigderson [174].…”

Section: Ramsey Theorymentioning

confidence: 99%

Gallai-Ramsey numbers for graphs and their generalizations

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This thesis contains research results on Gallai-Ramsey theory and its generalizations, which were obtained by the author with collaborators between September 2017 and February 2021. Apart from an introductory chapter (Chapter 1), the reader will find six closely related technical chapters (Chapters 2-7), which are mainly based on the research results that the author obtained when he was working as a PhD student at Northwestern Polytechnical University, Xi'an and the University of Twente.Chapters 2 and 3 focus on determining the exact values of Gallai-Ramsey numbers for several graphs. The other chapters mainly focus on studying various generalizations or variants of Gallai-Ramsey theory. In Chapter 4, we consider two extremal problems related to Gallai-colorings. In Chapter 5, we study the Erdős-Gyárfás function with respect to Gallai-colorings. In Chapter 6, we present a forbidden rainbow subgraph condition for an edge-colored graph to have a highly-connected monochromatic subgraph. In Chapter 7, we deal with the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. Papers underlying this thesis[1] Gallai-Ramsey numbers for a class of graphs with five vertices, Graphs and Combinatorics 36 (2020), 1603-1618 (with L. Wang). (Chapter 2) [2] Extremal problems and results related to Gallai-colorings, Discrete Mathematics 344 (2021), 112567 (with H.J. Broersma and L. Wang). (Chapters 3 and 4) vii viii Preface [3] The Erdős-Gyárfás function with respect to Gallai-colorings, submitted (with H.J. Broersma and L. Wang). (Chapter 5) [4] Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs, Discrete Applied Mathematics 285 (2020), 18-27 (with L. Wang).(Chapter 6)[5] Integer colorings with no rainbow 3-term arithmetic progression, submitted (with H.J. Broersma and L. Wang).

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An improved lower bound on multicolor Ramsey numbers

Cited by 15 publications

References 10 publications

An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

A note on multicolor Ramsey number of small odd cycles versus a large clique

Gallai-Ramsey numbers for graphs and their generalizations

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