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 $gr_{k}(K_{3} : H)$ is defined to be the minimum number of vertices $n$ for which any $k$-coloring of the complete graph $K_{n}$ 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

Density of Gallai Multigraphs

Diwan and Mubayi asked how many edges of each color could be included in a 33-edge-colored multigraph containing no rainbow triangle. We answer this question under the modest assumption that the multigraphs in question contain at least one edge between every pair of vertices. We also conjecture that this assumption is, in fact, without loss of generality