For two graphs $G$ and $H$, write $G \stackrel{ \mathrm{ rbw}}{ \longrightarrow} H$ 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 $G$ is an $n$-vertex graph with edge-density at least $d >0$, and $d$ is independent of $n$. In a companion article, we proved that the threshold for the property $G \cup \mathbb{ G}(n,p) \stackrel{ \mathrm{ rbw}}{ \longrightarrow} K_\ell$ is $n^{ -1/m_2(K_{ \left\lceil \ell/2 \right\rceil})}$, whenever $\ell \geq 9$. For smaller $\ell$, the thresholds behave more erratically, and for $4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for \emph{ large} cliques. In particular, we show that the thresholds for $\ell \in \{ 4, 5, 7\}$ are $n^{ -5/4}$, $n^{ -1}$, and $n^{ -7/15}$, respectively. For $\ell \in \{ 6, 8\}$ we determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are $n^{ -(2/3 + o(1))}$ and $n^{ -(2/5 + o(1))}$, respectively. For $\ell = 3$, the threshold is $n^{ -2}$; this follows from a more general result about odd cycles in our companion paper.
Categories: math.CO