Large rainbow cliques in randomly perturbed dense graphs

For two graphs G and H, write G rbw Shoham Letzter † −→ H if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G∪G(n,p), where G is an n-vertex graph with edge-density at least d, and d is a constant that does not depend on n. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property G ∪ G(n,p) rbw −→Ks for every s. In this paper, we show that for s ≥ 9 the threshold is n−1/m2(K⌈s/2⌉); in fact, our 1-statement is a supersaturation result. This turns out to (almost) be the threshold for s = 8 as well, but for every 4 ≤ s ≤ 7, the threshold is lower; see our companion paper for more details. In this paper, we also consider the property G∪G(n,p) rbw −→C2ℓ−1, and show that the threshold for this property is n−2 for every ℓ ≥ 2; in particular, it does not depend on the length of the cycle C2ℓ−1. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.