Small rainbow cliques in randomly perturbed dense graphs

For two graphs G and H, write G rbw -& RARR; 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 boolean OR G(n, p), where G is an n-vertex graph with edge-density at least d > 0, and d is independent of n. In a companion paper, we proved that the threshold for the property G boolean OR G(n, p) -& RARR;rbw Kl is n-1/m2(K left ceiling l/2 right ceiling ), whenever l & GE;9. For smaller l, the thresholds behave more erratically, and for 4 & LE; l & LE; 7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques. In particular, we show that the thresholds for l & ISIN; {4, 5, 7} are n-5/4, n-1, and n-7/15, respectively. For l & ISIN; {6, 8} we determine the threshold up to a (1 + o(1))-factor in the exponent: they are n-(2/3+o(1)) and n-(2/5+o(1)), respectively. For l = 3, the threshold is n-2; this follows from a more general result about odd cycles in our companion paper. (c) 2021 Elsevier Ltd. All rights reserved.