2021
DOI: 10.48550/arxiv.2105.02383
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Ramsey numbers of sparse digraphs

Abstract: Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr-Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If H is an acyclic digraph, the oriented Ramsey number of H, denoted − → r1 (H), is the least N such that every tournament on N vertices contains a copy of H. We show that for any ∆ ≥ 2 and any su… Show more

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Cited by 2 publications
(3 citation statements)
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“…As we proved (6), we can use the above to obtain that for uv ∈ E(G), P 2 (u, v) < k 2 + ℓ and P 2 (v, u) < k 2 + ℓ. (8) Moreover, as we proved Claim 2, we can show that any vertex v satisfies |B r G (v)| ≤ 10r(k 2 +ℓ) for any r ∈ N. In particular, with r = 1, this implies that any vertex v has degree at most 10(k 2 + ℓ) in graph G. By Turan's theorem, G contains an independent set of size at least n 10(k 2 +ℓ) ≥ n 10(k 2 +n/(10k)−k 2 ) = k. Take such an independent set {v 1 , . .…”
Section: Proofs Of the Theoremsmentioning
confidence: 71%
See 1 more Smart Citation
“…As we proved (6), we can use the above to obtain that for uv ∈ E(G), P 2 (u, v) < k 2 + ℓ and P 2 (v, u) < k 2 + ℓ. (8) Moreover, as we proved Claim 2, we can show that any vertex v satisfies |B r G (v)| ≤ 10r(k 2 +ℓ) for any r ∈ N. In particular, with r = 1, this implies that any vertex v has degree at most 10(k 2 + ℓ) in graph G. By Turan's theorem, G contains an independent set of size at least n 10(k 2 +ℓ) ≥ n 10(k 2 +n/(10k)−k 2 ) = k. Take such an independent set {v 1 , . .…”
Section: Proofs Of the Theoremsmentioning
confidence: 71%
“…In other words, this asks if degenerate graphs has linear (or almost linear) Ramsey number. This was disproved by Fox, He, and Wigderson [8]. They showed that for each ∆ ≥ 2 there exists an acyclic digraph H with maximum degree max v (d…”
Section: Introductionmentioning
confidence: 98%
“…In other words, is it the case that, for any positive integer d, there is a constant C(d) > 0 such that for any acyclic digraph D on n vertices with maximum degree d, every tournament on at least C(d)n vertices contains a copy of D? Very recently Fox, He and Widgerson [8] gave a negative answer to this question. Indeed, they showed that for all ∆ ≥ 2 and every sufficiently large n, there is an acyclic digraph D on n vertices with maximum degree ∆ for which there are tournaments on at least n Ω(∆ 2/3−o (1) ) vertices that do not contain any copy of D.…”
Section: Introductionmentioning
confidence: 99%