Abstract. We study how many copies of a graph [Formula: see text] that another graph [Formula: see text] with a given number of cliques is guaranteed to have. For example, one of our main results states that for all [Formula: see text], if [Formula: see text] is an [Formula: see text]-vertex graph with [Formula: see text] triangles and [Formula: see text] is sufficiently large in terms of [Formula: see text], then [Formula: see text] contains at least [Formula: see text] copies of [Formula: see text], and, furthermore, we show that these bounds are essentially best possible provided that either [Formula: see text] or certain bipartite analogues of well-known conjectures for Turán numbers hold.

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.