On an anti-Ramsey threshold for random graphs

For graphs G and H, let G⟶prbH denote the property that, for every proper edge-colouring of G (with an arbitrary number of colours) there is a totally multicoloured, or rainbow, copy of H in G, that is, a copy of H with no two edges of the same colour. We consider the problem of establishing the threshold pHrb=pHrb(n) of this property for the binomial random graph G(n,p). More specifically, we give an upper bound for pHrb and we extend our result to certain locally bounded colourings that generalize proper colourings. Our method is heavily based on a characterization of sparse quasi-randomness given by Chung and Graham (2008).