SIZE RAMSEY NUMBERS INVOLVING MATCHINGS

Abstract

Let F, G and H be finite, simple and undirected graphs. The edges and number of edges of a graph F will be denoted by E(F) and |E(F)| respectively. A graph F→ (G,H) if every 2-coloring (say red and blue) of E(F) produces either a “red” G or a “blue” H. The size Ramsey number r(G, H) = min{|E(F)|: F → (G,H)}. For t ≥ 1, the graph consisting of t independent edges will be denoted by tK2. In this paper, bounds and in some cases exact values will be calculated for r(tK2, G) for various classical graphs G, for example, when G is either a small order graph, a path, a cycle, a complete graph or a complete bipartite graph. Asymptotic results are obtained for some graphs in which exact values could not be calculated.