A General Lower Bound on Gallai-Ramsey Numbers for Non-Bipartite Graphs

Given a graph H and a positive integer k, the k-color Gallai-Ramsey number grk(K3 : H) is defined to be the minimum number of vertices n for which any k-coloring of the complete graph Kn contains either a rainbow triangle or a monochromatic copy of H. The behavior of these numbers is rather well understood when H is bipartite but when H is not bipartite, this behavior is a bit more complicated. In this short note, we improve upon existing lower bounds for non-bipartite graphs H to a value that we conjecture to be sharp up to a constant multiple