Ramsey number of a cycle versus a graph of a given size

TL;DR

This paper establishes an upper bound on the Ramsey number for a cycle versus a graph with given edges, solving a longstanding problem in combinatorics.

Contribution

It proves a new upper bound on the Ramsey number for cycles versus graphs with many edges, resolving a problem posed by Erdős et al.

Findings

The Ramsey number R(C_k,H) is at most 2m + floor((k-1)/2) for large m.

The result applies to graphs with no isolated vertices.

It confirms a conjecture for sufficiently large graphs.

Abstract

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $2m+\lfloor \frac{k-1}{2} \rfloor$, provided $m$ is sufficiently large with respect to $k$. This settles a problem of Erd\H{o}s, Faudree, Rousseau and Schelp.