Small rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{$\mathrm{r}\mathrm{b}\mathrm{w}$}{\longrightarrow} H$if$G$ has the property that every \emph{proper} colouring of its edges yields a \emph{rainbow} copy of $H$. We study the thresholds for such so-called \emph{anti-Ramsey} properties in randomly perturbed dense graphs, which are unions of the form $G \cup $\mathbb{G}$(n,p)$,where$G$isan$n$-vertexgraphwithedge-densityatleast$d >0$,and$d$isindependentof$n$. In a companion article, we proved that the threshold for the property $G\cup \mathbb{G}(n,p)K_{\mathrm{\ell }}$ is $n^{-1\mathrm{/}{m}_2(K_{\lceil \mathrm{\ell }\mathrm{/}2\rceil })}$, whenever $\ell \geq 9$\mathrm{.}Forsmaller$\ell$,thethresholdsbehavemoreerratically,andfor$4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for \emph{large} cliques. In particular, we show that the thresholds for $\mathrm{\ell }\in \{4,5,7\}$ are $n^{-5\mathrm{/}4}$, $n^{-1}$, and $n^{-7\mathrm{/}15}$, respectively. For $\mathrm{\ell }\in \{6,8\}$ we determine the threshold up to a $(1+o(1))$-factor in the exponent: they are $n^{-(2\mathrm{/}3+o(1))}$ and $n^{-(2\mathrm{/}5+o(1))}$, respectively. For $\mathrm{\ell }=3$, the threshold is $n^{-2}$; this follows from a more general result about odd cycles in our companion paper.