Gallai–Ramsey numbers for multiple triangles

Abstract

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses at most k colors. Given an integer k≥2, and graphs H1,H2,…,Hk, the Gallai-Ramsey number GR(H1,H2,…,Hk) is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of Hi in color i for some i∈{1,2,…,k}. Let mG denote the union of m disjoint copies of G. This paper studies the Gallai–Ramsey number GR(t1K3,…,tkK3) for all integers t1≥t2≥⋯≥tk≥1. The case t1=t2=⋯=tk=1 was obtained by Chung and Graham (1983). In this paper, we obtain a general bound of GR(t1K3,…,tkK3) for all t1≥t2≥⋯≥tk≥1, and prove that both the upper and lower bounds can be attained for some ti, i∈{1,…,k}.