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

“…n ˆ?n grid, an important result of Kohayakawa, Rödl, Szemerédi, and Schacht [17], which says that every graph H with n vertices and maximum degree ∆ satisfies rpHq ď n 2´1{∆`op1q , immediately yields the bound n 7{4`op1q . This was recently improved by Clemens, Miralaei, Reding, Schacht, and Taraz [6] to n 3{2`op1q (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.…”

Three early problems on size Ramsey numbers

“…The fourth question, about the size Ramsey number of paths, was resolved by Beck [2], who proved the surprising result that r(P n ) = Θ(n) for the path P n with n vertices. This breakthrough inspired many of the subsequent developments in the field, such as the classic papers [3,17,20,23,28] and the more recent results in [4,5,6,12,13,14,18,19,22,25].…”