Small rainbow cliques in randomly perturbed dense graphs

For two graphs G and H, write G⟶rbwH 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)⟶rbwK_ℓ is n^{−1/m₂(K_ℓ/2)}, whenever ℓ≥9. For smaller ℓ, the thresholds behave more erratically, and for 4≤ℓ≤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∈{4,5,7} are n^−5/4, n⁻¹, and n^−7/15, respectively. For ℓ∈{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.