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  p H rb = p H rb( n )of this property for the binomial random graph  G ( n , p ) is asymptotically at most  n − 1 / m ( 2 )( H ), wherem ( 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, thenp H 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 ≥ D n − βfor 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,p F rb ≪ n − 1 / m ( 2 )( F )for any such graph  F .