Monochromatic components with many edges

TL;DR

This paper investigates the maximum size of monochromatic connected components in edge-colored complete graphs, providing a complete solution for the case of four colors and establishing optimal bounds.

Contribution

The authors introduce a general framework for analyzing monochromatic components and fully resolve the case of four colors, including optimal bounds and constructions.

Findings

Any 4-coloring of K_n contains a monochromatic component with at least 1/12 of the edges

The 1/12 bound is proven to be optimal for certain colorings

The paper extends previous results for 2 and 3 colors to the case of 4 colors

Abstract

Given an $r$-edge-coloring of the complete graph $K_n$, what is the largest number of edges in a monochromatic connected component? This natural question has only recently received the attention it deserves, with work by two disjoint subsets of the authors resolving it for the first two special cases, when $r = 2$ or $3$. Here we introduce a general framework for studying this problem and apply it to fully resolve the $r = 4$ case, showing that any $4$-edge-coloring of $K_n$ contains a monochromatic component with at least $\frac{1}{12}\binom{n}{2}$ edges, where the constant $\frac{1}{12}$ is optimal only when the coloring matches a certain construction of Gy\'arf\'as.