Stars versus stripes Ramsey numbers

Abstract

For given simple graphs $G_1,G_2,\dots ,G_t$, the Ramsey number $R(G_1,G_2,\dots ,G_t)$ is the smallest positive integer $n$ such that if the edges of the complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,\dots ,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, for positive integers $t_1,t_2,\dots ,t_s$ and $n_1,n_2,\dots ,n_c$ the Ramsey number $R(S_{t_1},S_{t_2},\dots ,S_{t_s},n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)$ is computed, where $n{K}_2$ denotes a matching (stripe) of size $n$, i.e., $n$ pairwise disjoint edges and $S_n$ is a star with $n$ edges. This result generalizes and strengthens significantly a well-known result of Cockayne and Lorimer and also a known result of Gyárfás and Sárközy.