Multipartite Ramsey Numbers of Union of Stars

Abstract

Let s and k be positive integers with k ≥ 2 and G 1 , G 2 , …, G k be simple graphs. The set multipartite Ramsey number , denoted by M s ( G 1 , G 2 , …, G k ) , is the smallest positive integer c such that any k -coloring of the edges of K c × s contains a monochromatic copy of G i in color i for some i ∈ {1, 2, …, k } . The size multipartite Ramsey number , denoted by m c ( G 1 , G 2 , …, G k ) , is the smallest positive integer s such that any k -coloring of the edges of K c × s contains a monochromatic copy of G i in color i for some i ∈ {1, 2, …, k } . In this paper, we establish some lower and upper bounds, and some exact values of multipartite Ramsey numbers for the union of stars.