Set and size multipartite Ramsey numbers for stars

Abstract

The set multipartite Ramsey number for stars Ms(K1,n1,…,K1,nk) is the smallest positive integer c such that any k-coloring of the edges of Kc×s contains a monochromatic star K1,ni in color i for some i, 1≤i≤k, where Kc×s denotes the complete multipartite graph having c classes with s vertices per each class. On the other hand, the size multipartite Ramsey number for stars, denoted by mc(K1,n1,…,K1,nk), is the smallest positive integer s such that any k-coloring of the edges of Kc×s contains a monochromatic copy of K1,ni in color i for some i, 1≤i≤k. In this note we compute both Ms(K1,n1,…,K1,nk) and mc(K1,n1,…,K1,nk), extending well-known results on the classical and the bipartite Ramsey numbers for stars, respectively.