Let 𝒜 be a set of subsets of the natural numbers, and let G_𝒜(n) be the maximum cardinality of a subset of {1, 2, . . . , n} that does not have any subsets that are in 𝒜. We consider the general problem of giving upper bounds on G_𝒜(n), and give new results for some 𝒜 that are closed under dilation. We specifically address some examples, including sets that do not contain geometric progressions of length k with integer ratio, sets that do not contain geometric progressions of length k with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., sets of the form {a, ar, as, ars}.