On the multicolor size Ramsey number of stars and cliques

Abstract

For given graphs H1,…,Ht, we say that G is Ramsey for H1,…,Ht and we write G⟶(H1,…,Ht), if no matter how one colors the edges of G with t colors, say 1,…,t, there exists a monochromatic copy of Hi in the ith color for some 1≤i≤t. The multicolor Ramsey number r(H1,…,Ht) is the smallest integer n such that the complete graph Kn is Ramsey for (H1,…,Ht). The multicolor size Ramsey number rˆ(H1,…,Ht) is defined as min{|E(G)|:G⟶(H1,…,Ht)}, while the restricted size Ramsey number rˆ∗(H1,…,Ht) is defined as rˆ∗(H1,…,Ht)=min{|E(G)|:G⟶(H1,…,Ht)and|V(G)|=r(H1,…,Ht)}.In 1978 Erdős et al. initiated the study of the size Ramsey numbers of graphs. Afterwards, size Ramsey numbers have been studied with particular focus on the case of trees, sparse graphs and bounded degree graphs. In this paper, the order of magnitude of multicolor size Ramsey number of stars and cliques is determined in terms of r(K1,q1,…,K1,qn) and r(Kp1,…,Kpm). Moreover, in some special cases, restricted size Ramsey number of stars and cliques is determined exactly. Our results have, up to a constant factor, similar order of magnitude and generalize significantly a well known result of Faudree and Sheehan.