On the maximal anti-Ramsey problem of Burr, Erdős, Graham, and Sós for $P_4$

Given a graph $L$, the maximal anti-Ramsey function $\text{chiS}(n,e,L)$ denotes the minimum integer $\text{chiS}$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $\text{chiS}$ colors in which each copy of $L$ in $G$ is rainbow. In 1989, Burr, Erdős, Graham, and Sós posed the following problem: Is it true that for all $\epsilon >0$, there exists $c(\epsilon )>0$ such that for all sufficiently large $n$, $\text{chiS}(n,\left(\genfrac{}{}{0px}{}{n}{2}\right)-\lfloor n^{2-\epsilon }\rfloor ,P_4)>c(\epsilon )n^2\mathrm{.}$ Very recently, Li, Ning, and Xie gave a negative answer to the problem for all $0<\epsilon <1\mathrm{/}2$. In this note, we establish that a quadratic lower bound holds in the complementary regime $\epsilon \ge 1\mathrm{/}2$. More specifically, we prove that for all $\epsilon \ge 1\mathrm{/}2$ and sufficiently large $n$, there is an absolute constant $c>0$ such that $\text{chiS}(n,\left(\genfrac{}{}{0px}{}{n}{2}\right)-\lfloor n^{2-\epsilon }\rfloor ,P_4)>c{n}^2\mathrm{.}$