For graphs G and H, let G rb ÝÑ p H denote the property that for every proper edge-colouring of G there is a rainbow H in G. It is known that, for every graph H, an asymptotic upper bound for the threshold function p rb H " p rb H pnq of this property for the random graph Gpn, pq is n´1 {m p2q pHq , where m p2q pHq denotes the so-called maximum 2-density of H. Extending a result of Nenadov, Person, Škorić, and Steger [J. Combin. Theory Ser. B 124 (2017), 1-38] we prove a matching lower bound for p rb K k for k ě 5. Furthermore, we show that p rb K4 " n´7 {15 . §1. IntroductionLet r be a positive integer and let G and H be graphs. We denote by G Ñ pHq r the property that any colouring of the edges of G with at most r colours contains a monochromatic H in G. In 1995, Rödl and Ruciński determined the threshold for the property Gpn, pq Ñ pHq r for all graphs H. The maximum 2-density m p2q pHq of a graph H is given bywhere we suppose |V pHq| ě 3.Theorem 1 (Rödl and Ruciński [8,9]). Let H be a graph containing a cycle. Then, the threshold function p H " p H pnq for the property Gpn, pq Ñ pHq r is given by p H pnq " n´1 {m p2q pHq .

For graphs G and H, let G→normalp rb H denote the property that for every proper edge‐coloring of G (with an arbitrary number of colors) there is a rainbow copy of H in G, that is, a copy of H with no two edges of the same color. The authors (2014) proved that, for every graph H, the threshold function pH rb =pH rb false(nfalse) of this property for the binomial random graph G(n,p) is asymptotically at most n−1/mfalse(2false)(H), where m(2)false(Hfalse) denotes the so‐called maximum 2‐density of H. Nenadov et al. (2014) proved that if H is a cycle with at least  seven vertices or a complete graph with at least 19 vertices, then pH rb =n−1/m(2)false(Hfalse). We show that there exists a fairly rich, infinite family of graphs F containing a triangle such that if p≥Dn−β for suitable constants D=D(F)>0 and β=β(F), where β>1/m(2)false(Ffalse), then Gfalse(n,pfalse)→normalp rb F almost surely. In particular, pF rb ≪n−1/m(2)false(Ffalse) for any such graph F.

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