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arxiv 2207.02920 v1 pith:ZRVMDCH4 submitted 2022-07-06 math.CO

The ErdH{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 , Pawe{\l} Pra{\l}at This is my paper
classification math.CO
keywords processcoloringcolorsfracusesanalyzeanalyzedbohman
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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 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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  1. New results on the odd- and unique-Ramsey numbers

    math.CO 2026-05 unverdicted novelty 6.0

    New lower bounds r_odd(n, K_{s,t}) > n^{1/(s/2 + 1/(2 floor(t/8)))} for odd s even t, r_u(n, C_n) > n/4 creating a polynomial gap, and odd-Ramsey number of Hamilton cycles >1 in super-Dirac graphs.