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 sufficiently large n, there exists an acyclic digraph H with n vertices and maximum degree ∆ such thatThis proves that − → r1 (H) is not always linear in the number of vertices for bounded-degree H. On the other hand, we show that − → r1 (H) is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs H, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs.For multiple colors, we prove a quasipolynomial upper boundfor all boundeddegree acyclic digraphs H on n vertices, where − → r k (H) is the least N such that every k-edge-colored tournament on N vertices contains a monochromatic copy of H. For k ≥ 2 and n ≥ 4, we exhibit an acyclic digraph H with n vertices and maximum degree 3 such that − → r k (H) ≥ n Ω(log n/ log log n) , showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices.

On 1-subdivisions of transitive tournaments

“…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 , . .…”

“…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…”

Powers of paths and cycles in tournaments

“…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.…”