For two graphs G and H, write G⟶rbw​H if G has the property that every {\sl proper} colouring of its edges
yields a {\sl rainbow} copy of H.
We study the thresholds for such so-called {\sl 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.Comment: 21 pages; some typos fixed in the last versio