The size-Ramsey number of cubic graphs

TL;DR

This paper proves an improved upper bound on the size-Ramsey number of cubic graphs, showing that a random graph with linear size in vertices is with high probability Ramsey for any cubic graph, refining previous bounds.

Contribution

The paper establishes a tighter upper bound of O(n^{8/5}) for the size-Ramsey number of cubic graphs, improving upon previous bounds and demonstrating the effectiveness of random graphs in Ramsey properties.

Findings

Size-Ramsey number of cubic graphs is O(n^{8/5})

Random graphs with linear size are Ramsey for cubic graphs with high probability

The bound on the number of vertices in the random graph is essentially optimal

Abstract

We show that the size-Ramsey number of any cubic graph with $n$ vertices is $O(n^{8/5})$, improving a bound of $n^{5/3 + o(1)}$ due to Kohayakawa, R\"{o}dl, Schacht, and Szemer\'{e}di. The heart of the argument is to show that there is a constant $C$ such that a random graph with $C n$ vertices where every edge is chosen independently with probability $p \geq C n^{-2/5}$ is with high probability Ramsey for any cubic graph with $n$ vertices. This latter result is best possible up to the constant.