The Erd\H{o}s-Gy\'arf\'as function $f(n, 4, 5) = \frac 56 n + o(n)$ -- so Gy\'arf\'as was right

TL;DR

This paper proves that the minimal number of colors needed for a specific edge-coloring of complete graphs grows asymptotically as 5/6 of the number of vertices, confirming a long-standing conjecture.

Contribution

It provides a probabilistic construction and analysis that settles a historical disagreement about the Erdős-Gyárfás function for (4,5)-colorings.

Findings

Existence of (4,5)-colorings with approximately 5/6 n colors

Use of a randomized triangle removal process for construction

Application of differential equation method for analysis

Abstract

A $(4, 5)$-coloring of $K_n$ is an edge-coloring of $K_n$ where every $4$-clique spans at least five colors. We show that there exist $(4, 5)$-colorings of $K_n$ using $\frac 56 n + o(n)$ colors. This settles a disagreement between Erd\H{o}s and Gy\'arf\'as reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollob\'as and Erd\H{o}s, and analyzed by Bohman, Frieze and Lubetzky.