Gallai-Ramsey numbers of books

Abstract

For a graph G, the Ramsey number Rk(G) is defined as the minimum r such that any k-edge coloring of Kr contains a monochromatic G. A Gallai k-coloring is a k-edge coloring of a complete graph without rainbow triangle, where a rainbow triangle is that all three edges are colored by three colors. The Gallai-Ramsey number GRk(G) is defined as the minimum r such that any Gallai k-coloring of Kr contains a monochromatic G. Call Bn book that is formed by n triangles sharing a common edge. For all k≥2 and n≥1, we show that GRk(Bn) is at most 1.1(4n+1)5k−22 if k is even and 2.75(4n+1)5k−32 if k is odd, respectively.