The threshold conjecture for rainbow even cliques in randomly perturbed dense graphs

Let $d>0$ be fixed, let $r\geq 5$ be an integer, let $\mathscr{G}_{d,n}$ denote an $n$-vertex graph of edge-density $d$, and let $\mathbb{G}(n,p)$ be the binomial random graph. Write $G\stackrel{\mathrm{rbw}}{\longrightarrow}K_s$ when every proper edge-colouring of $G$ contains a rainbow copy of $K_s$.

Even-clique threshold conjecture. For every real $d>0$ and every integer $r\geq 5$, the threshold for the property

$$\mathscr{G}_{d,n}\cup\mathbb{G}(n,p)\stackrel{\mathrm{rbw}}{\longrightarrow}K_{2r}$$

is $n^{-1/m_2(K_r)}$.

The conjecture predicts that the lower bound obtained from proper colourings of the random perturbation is sharp for even complete graphs. The paper proves an upper bound at the larger scale $n^{-(r-2)/\binom{r}{2}}$, leaving a gap; the exact threshold remains open.