On an anti-Ramsey property of random graphs

For graphs G and H, let G → p m c H denote the property that, for every proper edge-colouring of G, with an arbitrary number of colours, there is a 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 p H m c = p H m c ( n ) of this property for the binomial random graph G ( n , p ) .