LARGE RAINBOW CLIQUES IN RANDOMLY PERTURBED DENSE GRAPHS

For two graphs G and H, write (Equation presented). H if G has the property that every proper coloring 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 (Equation presented). In this paper, we show that for s ≥ 9 the threshold is (Equation presented). 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. Also in this paper, we determine that the threshold for the property (Equation presented). for every l ≤ 2; in particular, the threshold does not depend on the length of the cycle C_{2l - 1}. For even cycles, and in fact any fixed bipartite graph, no random edges are needed at all; that is, (Equation presented). always holds, whenever G is as above and H is bipartite.