The size‐Ramsey number of cubic graphs

Abstract

We show that the size-Ramsey number of any cubic graph with n vertices is Opn 8{5 q, improving a bound of n 5{3`op1q due to Kohayakawa, Rödl, Schacht, and Szemerédi. The heart of the argument is to show that there is a constant C such that a random graph with Cn vertices where every edge is chosen independently with probability p ě Cn ´2{5 is with high probability Ramsey for any cubic graph with n vertices. This latter result is best possible up to the constant.

On the size-Ramsey number of grids

“…Regarding upper bounds for the size-Ramsey number of the √ n × √ n grid, an important result of Kohayakawa, Rödl, Schacht, and Szemerédi [17], which says that every graph H with n vertices and maximum degree satisfies r(H) n 2−1/ +o (1) , immediately yields the bound n 7/4+o (1) . This was recently improved by Clemens, Miralaei, Reding, Schacht, and Taraz [6] to n 3/2+o (1) (and an alternative proof of this bound was also noted in our recent paper [7]). The goal of this short note is to provide an elementary proof of an improved upper bound.…”

On bounded degree graphs with large size-Ramsey numbers

“…We refer to papers [2,3] for improvements of the upper bound as well as further references regarding size-Ramsey numbers. Whereas there has been substantial progress on estimating the size-Ramsey numbers of sparse graphs from above, the lower bound cn log 1 60 n given in [6] seems to be the best known estimate as of this writing.…”

The Turán number of the grid

“…Clemens, Miralaei, Reding, Schacht and Taraz [5] gave an upper bound for the size Ramsey number of the $$t\times t$$ grid. Their bound was improved very recently by Conlon, Nenadov and Trujić [6]. Kim, Lee and Lee [17] proved Sidorenko's conjecture for grids (in arbitrary dimension).…”