Small rainbow cliques in randomly perturbed dense graphs

Abstract

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 > 0, and d is independent of n.In a companion paper, we proved that the threshold for the property G ∪ G(n, p)) , whenever ≥ 9. For smaller , the thresholds behave more erratically, and for 4 ≤ ≤ 7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques.In particular, we show that the thresholds for ∈ {4, 5, 7} are n −5/4 , n −1 , and n −7/15 , respectively. For ∈ {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 = 3, the threshold is n −2 ; this follows from a more general result about odd cycles in our companion paper.