Gallai–Ramsey numbers of [formula omitted] with multiple colors

Abstract

We study Ramsey-type problems in Gallai-colorings. Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) is the least positive integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. It turns out that GRk(H) behaves more nicely than the classic Ramsey number Rk(H). However, finding exact values of GRk(H) is far from trivial. In this paper, we prove that GRk(C7)=3⋅2k+1 for all k≥1. Our technique relies heavily on the structural result of Gallai on edge-colorings of complete graphs without rainbow triangles.