Anti-Ramsey properties of random graphs

We call a coloring of the edge set of a graph G a b-bounded coloring if no color is used more than b times. We say that a subset of the edges of G is rainbow if each edge is of a different color. A graph has property A ( b , H ) if every b-bounded coloring of its edges has a rainbow copy of H. We estimate the threshold for the random graph G n , p to have property A ( b , H ) .