A note on pseudorandom Ramsey graphs

Mubayi, Dhruv; Verstraëte, Jacques

doi:10.48550/arxiv.1909.01461

Cited by 13 publications

(13 citation statements)

References 31 publications

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“…However, it is conjectured that such a graph exists. If so, the proof would carry over and we would obtain the following conjecture, which would show that Theorem 1.1 is tight up to a constant function of r for sufficiently large n. It is known that other interesting results would follow from knowing the existence of such pseudorandom K r -free graphs, including giving nearly tight bounds for off-diagonal Ramsey numbers (see [33,38]).…”

Section: Discussionmentioning

confidence: 96%

Making an $H$-Free Graph $k$-Colorable

Fox¹,

Zoe²,

Mani³

2021

Preprint

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We study the following question: how few edges can we delete from any H-free graph on n vertices in order to make the resulting graph k-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H any fixed odd cycle, we determine the answer up to a constant factor when n is sufficiently large. We also prove an upper bound when H is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.

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Section: Discussionmentioning

confidence: 96%

Making an $H$-Free Graph $k$-Colorable

Fox¹,

Zoe²,

Mani³

2021

Preprint

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“…Sudakov, Szabó and Vu [22] raised the question whether, for fixed r ≥ 3, there exist K r -free (n, d, λ)-graphs with d = Ω(n 1−1/(2r−3) ) and λ = O( √ d). It has been proved that this would have striking consequences for off-diagonal Ramsey numbers [18]. Speculating that such graphs do exist, this would via (1.4)…”

Section: Conjecture 14 ([6]mentioning

confidence: 99%

New results for MaxCut in $H$-free graphs

Glock,

Janzer,

Sudakov

2021

Preprint

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The MaxCut problem asks for the size mc(G) of a largest cut in a graph G. It is well known that mc(G) ≥ m/2 for any m-edge graph G, and the difference mc(G) − m/2 is called the surplus of G. The study of the surplus of H-free graphs was initiated by Erdős and Lovász in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is Ω(m 4/5 ), and found constructions matching this bound. We prove several new results in this area.(i) We show that for every fixed odd r ≥ 3, any C r -free graph with m edges has surplus Ω r m r+1 r+2 . This is tight, as is shown by a construction of pseudorandom C r -free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when r is even.(ii) Generalizing the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs our bounds on the surplus are sharp.(iii) We prove that an n-vertex graph with few copies of K r and average degree d has surplus Ω r (d r−1 /n r−3 ), which is tight when d is close to n provided that a conjectured dense pseudorandom K r -free graph exists. This result is used to improve the best known lower bound (as a function of m) on the surplus of K r -free graphs.Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

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“…It is known (using the expander mixing lemma, see for example [AS16, Corollary 9.2.5]) that the edge density of an optimally pseudorandom K k -free graph is O(n −1/(2k−3) ). If optimally pseudorandom graphs with this edge density exist, it follows from the results of Mubayi and Verstraëte [MV19], and the multicolour generalization by He and Wigderson [HW20], that the determination of the off-diagonal Ramsey number r(k 1 , . .…”

Section: Introductionmentioning

confidence: 95%