The proposed sharp threshold for rainbow connection number at least three

Let $r$ be a fixed integer with $r\geqslant 3$. For a graph $G$, let $\mathrm{rc}(G)$ denote its rainbow connection number, and define

$$\mathcal{R}_r=\\{G:\mathrm{rc}(G)\leqslant r\\}.$$

Set

$$C=\frac{r^{r-2}}{(r-2)!},\qquad p(n)=\frac{\left(C\log n\right)^{1/r}}{n^{1-1/r}}.$$

Proposed threshold conjecture. The function $p(n)$ is a sharp threshold for the graph property $\mathcal{R}_r$.

This proposes that, for fixed $r\geqslant 3$, the threshold for rainbow connection number at most $r$ differs from the threshold for diameter at most $r$, unlike the known case $r=2$. The paper proves that this function is an upper bound for the threshold, while the asserted sharp-threshold statement is not resolved in the supplied text.