On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós

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, Erdős, Graham, and Sós conjectured that for every odd cycle $C_{2k+1}$ with $k\ge 3$, $f(n,\lfloor n^2\mathrm{/}4\rfloor +1,C_{2k+1})=n^2\mathrm{/}8+o(n^2)$. In this note, we confirm this conjecture for all $k\ge 4$. More generally, we establish the asymptotic formula $$f(n,e,C_{2k+1})=\frac{e}{2}+\frac{n}{2}\sqrt{e-\frac{n^2}{4}}+o(n^2),$$ for the entire non-trivial range of $\lfloor n^2\mathrm{/}4\rfloor +1\le e\le \left(\genfrac{}{}{0px}{}{n}{2}\right)$.