A $(5,5)$-coloring of $K_n$ with few colors

TL;DR

This paper constructs an explicit edge-coloring of complete graphs that significantly reduces the number of colors needed to prevent small monochromatic cliques, improving previous bounds and approaching theoretical limits.

Contribution

The authors provide a new explicit construction of a (5,5)-coloring of $K_n$ that improves the upper bound on the number of colors from $O(n^{1/2})$ to approximately $n^{1/3}$, nearing the lower bound.

Findings

Constructed an explicit (5,5)-coloring with $f(n,5,5) \, \leq \, n^{1/3 + o(1)}$

Improved the upper bound from probabilistic methods by a factor of roughly $\sqrt{n}$

Bridged the gap between known upper and lower bounds for $f(n,5,5)$.

Abstract

For fixed integers $p$ and $q$, let $f(n,p,q)$ denote the minimum number of colors needed to color all of the edges of the complete graph $K_n$ such that no clique of $p$ vertices spans fewer than $q$ distinct colors. Any edge-coloring with this property is known as a $(p,q)$-coloring. We construct an explicit $(5,5)$-coloring that shows that $f(n,5,5) \leq n^{1/3 + o(1)}$ as $n \rightarrow \infty$. This improves upon the best known probabilistic upper bound of $O\left(n^{1/2}\right)$ given by Erd\H{o}s and Gy\'{a}rf\'{a}s, and comes close to matching the best known lower bound $\Omega\left(n^{1/3}\right)$.