The anti-Ramsey threshold of complete graphs

Abstract

For graphs G and H, let G→rbH denote the property that, for every proper edge-colouring of G, there is a rainbow H in G. For every graph H, the threshold function pHrb=pHrb(n) of this property in the random graph G(n,p) satisfies pHrb=O(n−1/m(2)(H)), where m(2)(H) denotes the so-called maximum 2-density of H. Completing a result of Nenadov, Person, Škorić, and Steger [J. Combin. Theory Ser. B 124 (2017), 1–38], we prove a matching lower bound for pKkrb for k⩾5. Furthermore, we show that pK4rb=n−7/15≪n−1/m(2)(K4).