The Erdős-Gyárfás function $f(n, 4, 5) = \frac 56 n + o(n)$ -- so Gyárfás was right

Patrick Bennett, Ryan Cushman, Andrzej Dudek, Paweł Prałat

Published 2022-07-06Version 1

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.