Ramsey numbers for multiple copies of hypergraphs

Abstract

In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such $G$ and $H$, among them are any disjoint union of $k$-uniform hypergraphs involving loose paths, loose cycles, tight paths, tight cycles with a multiple of $k$ edges, stars, Kneser hypergraphs and complete $k$-uniform $k$-partite hypergraphs for $G$ and linear hypergraphs for $H$. As an application, $R(mG,nH)$ is determined where $m$ or $n$ is large and $G$ and $H$ are either loose paths, loose cycles, tight paths, or stars. Also, $R(G,nH)$ is determined when $G$ is a bipartite graph with a matching saturating one of its color classes and $H$ is an arbitrary graph for sufficiently large $n$. Moreover, some bounds are given for $R(mG,nH)$ which allow us to determine this Ramsey number when $m\geq n$ and $G$ and $H$, $(|V(G)|\geq |V(H)|)$, are 3-uniform loose paths or cycles, $k$-uniform loose paths or cycles with at most 4 edges and $k$-uniform stars with 3 edges.