2024/10/10 by McGuigan, Isabel, Pan, Katherine
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.07634
In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph Kn1, n2 be before any coloring of its edges with r colors contains either a monochromatic copy of G = Ks,t or a rainbow copy of H = Ks,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 Kn1,n2 and colorings of a subset of (n1+n2)-dimensional Euclidean space, we prove new upper bounds on n for some configurations which can be expressed as Cartesian products of simplices.