Ramsey numbers for the pair sparse graph-path or cycle

Introduction. Let G and H be simple graphs. The Ramsey number r(G, H) is the smallest integer n such that for each graph F on n vertices, either G is a subgraph of F or H is a subgraph of F, the complement of F. Calculation of r(G, H) for particular pairs of graphs G and H has received considerable attention, and a survey of such results can be found in [2]. Chvatal [5] proved that if Tn is a tree on n vertices and Km is a complete graph on m vertices, then r(T7,, Kin) = (n 1)(m 1) + 1. In [4] it was shown that if Tn is replaced by a sparse connected graph Gn on n vertices the Ramsey number remains the same (i.e. r(G,, Kin) = (n 1)(m 1) + 1). For m = 3 Chvatal's theorem implies r(T1, K3) = 2n -1. In this paper we will show that if Tn is replaced by any sparse connected graph G on n vertices and K3 is replaced by an odd cycle Ck, then for appropriate n the Ramsey number is unchanged. In particular we will prove the following.