Size-Ramsey numbers of graphs with maximum degree three

TL;DR

This paper improves the upper bound on the size-Ramsey number for graphs with maximum degree three from roughly n^{8/5} to n^{3/2+o(1)} using a novel host graph construction, advancing understanding in Ramsey theory.

Contribution

The paper introduces a new host graph construction to establish a tighter upper bound on size-Ramsey numbers for degree three graphs.

Findings

Bound on size-Ramsey number improved to n^{3/2+o(1)}

Novel host graph construction used instead of binomial random graphs

Results reach a natural barrier of existing methods

Abstract

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the upper bound of $n^{5/3 + o(1)}$ by Kohayakawa, R\"odl, Schacht and Szemer\'edi. In this paper we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.