On an anti‐Ramsey threshold for sparse graphs with one triangle

For graphs G and H, let G→p 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 (n) of this property for the binomial random graph G(n,p) is asymptotically at most n−1/m(2)(H), where m(2)(H) 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)(H). 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)(F), then G(n,p)→p rb F almost surely. In particular, pF rb ≪n−1/m(2)(F) for any such graph F.