The size Ramsey and connected size Ramsey numbers for matchings versus paths

Abstract

Given simple graphs F, G, and H, we write F → (G, H) if for every red-blue coloring of the edges of F, there exists either a red subgraph G or a blue subgraph H in F. The size Ramsey number for G and H, denoted by (G, H), is the smallest size of a graph F which satisfies F → (G, H). If in addition F must be connected, then the resulting number is called the connected size Ramsey number for G and H, denoted by (G, H). In this paper, we obtain upper bounds for , and for t ≥ 1 and m ≥ 3. We also determine the exact values of and . In addition, the exact values of for t = 3,4 are given.