Abstract. Let [Formula: see text] be the maximum number of copies of a graph [Formula: see text] in an [Formula: see text]-vertex graph which contains no copy of a graph from the graph family [Formula: see text]. We say that a rational number [Formula: see text] is realizable for [Formula: see text] if there exists a finite family [Formula: see text] such that [Formula: see text]. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between 1 and 2 is realizable for [Formula: see text]. We generalize their result to show that every rational between 1 and [Formula: see text] is realizable for [Formula: see text] for all [Formula: see text]. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem.