Gallai and [formula omitted]-uniform Ramsey numbers of complete bipartite graphs

Abstract

A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k, the Gallai Ramsey number GRk(H) is the minimum integer N such that every Gallai k-coloring of KN contains a monochromatic copy of H. A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and ℓ, the ℓ-uniform Ramsey number Rkℓ(H) is the minimum integer N such that any uniform k-coloring of any complete multipartite graph on N vertices with each part of cardinality no more than ℓ, contains a monochromatic copy of H. Let Km,n denote a complete bipartite graph. Wu et al. conjectured that GRk(Km,n)=(n−1)(k−2)+R21(Km,n) if R21(Km,n)≥3m−2, where k≥3 and m≥n≥2. In this paper, we show that GRk(Km,n)=(n−1)(k−2)+R2m−1(Km,n) for all integers k≥3 and m≥n≥2. Based on this result, we improve the known general bounds for GRk(Km,n), and confirm the conjecture for some complete bipartite graphs.