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arXiv · 2603.18952

On a maximal anti-Ramsey conjecture of Burr, Erd\H{o}s, Graham, and S\'os

Matija Bucic·Kaizhe Chen·Jie Ma
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Abstract

Given a graph $H$, the maximal anti-Ramsey function $f(n,e,H)$ denotes the minimum integer $f$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $f$ colors in which each copy of $H$ in $G$ is rainbow. In the late 1980s, Burr, Erd\H{o}s, Graham, and S\'os conjectured that for every odd cycle $C_{2k+1}$ with $k \ge 3$, $f(n, \lfloor n^2/4 \rfloor + 1, C_{2k+1}) = n^2/8 + o(n^2)$. In this note, we confirm this conjecture for all $k \ge 4$. More generally, we establish the asymptotic formula $$f\left(n,e,C_{2k+1}\right)=\frac{e}{2}+\frac{n}{2}\sqrt{e-\frac{n^2}{4}}+o(n^2),$$ for the entire non-trivial range of $\left\lfloor n^2/4 \right\rfloor+1\le e\le \binom{n}{2}$.

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Matija Bucic, Kaizhe Chen, Jie Ma. 2026-03-19. On a maximal anti-Ramsey conjecture of Burr, Erd\H{o}s, Graham, and S\'os. https://arxiv.org/abs/2603.18952

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