Rainbow Cliques in Randomly Perturbed Dense Graphs

For two graphs G and H, write G⟶ rbw 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. We determine the threshold for the property G∪ G(n, p) ⟶ rbw K_s for every s. 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 and is different for each 4 ≤ s≤ 7. Moreover, we prove that for every ℓ≥ 2 the threshold for the property G∪ G(n, p) ⟶ rbw C_{2 ℓ - 1} is n^{- 2} ; in particular, the threshold does not depend on the length of the cycle C_{2 ℓ - 1}. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.