Small rainbow cliques in randomly perturbed dense graphs

Elad Aigner-Horev, Oran Danon, Dan Hefetz, Shoham Letzter

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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 is n−1/m2(Kℓ/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.

Original languageEnglish
Article number103452
JournalEuropean Journal of Combinatorics
StatePublished - Mar 2022


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