Anti-Ramsey Threshold of Cycles for Sparse Graphs

For graphs G and H, let G ⟶ p rb H denote the property that for every proper edge colouring of G there is a rainbow copy of H in G. Extending a result of Nenadov, Person, Skoric and Steger (2017), we prove that n−1/m2(Cl) is the threshold for G ( n , p ) ⟶ p rb C l when l ≥ 5. Thus our result together with a result of the third author which says that the threshold for G ⟶ p rb C 4 is n−3/4 settles the problem of determining the threshold for G ⟶ p rb C l for all values of l.