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arXiv · 2207.02920

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

Patrick Bennett·Ryan Cushman·Andrzej Dudek·Paweł Prałat
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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.

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Patrick Bennett, Ryan Cushman, Andrzej Dudek, Paweł Prałat. 2022-07-06. The Erd\H{o}s-Gy\'arf\'as function $f(n, 4, 5) = \frac 56 n + o(n)$ -- so Gy\'arf\'as was right. https://arxiv.org/abs/2207.02920

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