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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 adisagreement 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-calleddifferential equation method to establish dynamic concentration. In particular,our coloring process uses random triangle removal, a process first introducedby Bollob\'as and Erd\H{o}s, and analyzed by Bohman, Frieze and Lubetzky.

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