Stars and Their k-Ramsey Numbers

Abstract

For bipartite graphs F and H with Ramsey number $R(F, H) = n$ and an integer k with $2 \le k \le n$, the k-Ramsey number of F and H is the minimum order of a balanced complete k-partite graph G for which every red-blue coloring of G results in a subgraph of G isomorphic to F all of whose edges are colored red or a subgraph isomorphic to H all of whose edges are colored blue. A formula is presented for the k-Ramsey number of every two stars $K_{1, s}$ and $K_{1, t}$ $(s, t \ge 2)$ and every integer k with $2 \le k \le R(K_{1, s}, K_{1, t})$.