Bipartite and Euclidean Gallai-Ramsey Theory

In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph $K_{n_1, n_2}$ be before any coloring of its edges with $r$ colors contains either a monochromatic copy of $G = K_{s,t}$ or a rainbow copy of $H = K_{s,t}$? We demonstrate that the answer is linear in $r$, and provide more precise bounds for the specific case $s = 2$. Furthermore, we also consider the following Euclidean Gallai-Ramsey question: given a configuration $H$ in Euclidean space, what is the smallest $n$ such that any $r$-coloring of $n$-dimensional Euclidean space contains a monochromatic or rainbow configuration congruent to $H$? Through a natural translation between edge colorings of the complete bipartite graph $K_{n_1,n_2}$ and colorings of a subset of $(n_1+n_2)$-dimensional Euclidean space, we prove new upper bounds on $n$ for some configurations which can be expressed as Cartesian products of simplices.