Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-k-coloring) is an edge-coloring (with colors from {1,2,…,k}) of a complete graph without rainbow triangles. Given a graph H and a positive integer k, the k-colored Gallai-Ramsey number GR_k(H) is the minimum integer n such that every Gallai-k-coloring of the complete graph K_n contains a monochromatic copy of H. In this paper, we consider two extremal problems related to Gallai-k-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a k-edge-coloring of K_n. Second, for n≥GR_k(K₃), we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-k-coloring of K_n, yielding the exact value for k=3. Furthermore, we determine the Gallai-Ramsey number GR_k(K₄+e) for the graph on five vertices consisting of a K₄ with a pendant edge.