Note on a generalization of Gallai-Ramsey numbers

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors appear on edges between the parts and at most one color appears on edges in between each pair of parts. We extend this notion by defining a coloring of a complete graph to be $k$-Gallai if every induced subgraph has a nontrivial partition of the vertices such that there are at most $k$ colors present in between parts of the partition. The generalized $(k, \ell)$ Gallai-Ramsey number of a graph $H$ is then defined to be the minimum number of vertices $N$ such that every $k$-Gallai coloring of a complete graph $K_{n}$ with $n \geq N$ using at most $\ell$ colors contains a monochromatic copy of $H$. We prove bounds on these generalized $(k, \ell)$ Gallai-Ramsey numbers based on the structure of $H$, extending recent results for Gallai colorings.