Bipartite and Euclidean Gallai-Ramsey Theory

TL;DR

This paper explores Gallai-Ramsey theory in bipartite graphs and Euclidean space, establishing linear bounds for complete bipartite graphs and new bounds for Euclidean configurations, advancing understanding of monochromatic and rainbow structures.

Contribution

It introduces linear bounds for Gallai-Ramsey numbers in bipartite graphs and extends these concepts to Euclidean space configurations, providing new upper bounds for specific geometric arrangements.

Findings

Gallai-Ramsey number for $K_{s,t}$ in bipartite graphs is linear in the number of colors.

Established bounds for Euclidean space configurations related to Cartesian products of simplices.

Connected bipartite graph colorings with Euclidean space configurations through a natural translation.

Abstract

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.