Minimizing Monochromatic Subgraphs of $K_{n,n}$

TL;DR

This paper investigates the size of the largest monochromatic subgraph in edge-colored complete bipartite graphs, providing bounds and constructions for various numbers of colors, with implications for understanding monochromatic structures.

Contribution

It offers new bounds and constructions for the maximum monochromatic subgraph size in $K_{n,n}$ for specific and general numbers of colors, advancing combinatorial understanding.

Findings

Bounds established for $r eq$ perfect square or one less than a perfect square.

A lower bound valid for all $r$, sharp for perfect squares.

A construction providing an upper bound, conjectured to be tight.

Abstract

Given any $r$-edge coloring of $K_{n,n}$, how large is the maximum (over all $r$ colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for $r \leq 8$, when $r$ is a perfect square, and when $r$ is one less than a perfect square all up to a constant additive term that depends on $r$. We give a lower bound on this quantity that holds for all $r$ and is sharp when $r$ is a perfect square up to a constant additive term that depends on $r$. Finally, we give a construction for all $r$ which provides an upper bound on this quantity up to a constant additive term that depends on $r$, and which we conjecture is also a lower bound.