In this talk we will study the so-called perturbed model which is a graph distribution 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 $n$.
We are interested in the threshold of the following anti-Ramsey property: every proper edge-colouring of $G \cup \mathbb{G}(n,p)$ yields a rainbow copy of $K_s$. We have determined this threshold for every $s$.
Based on joint work with Elad Aigner-Horev, Oran Danon and Shoham Letzter.