Sets of natural numbers with proscribed subsets

Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give some new upper bounds on some families that are closed under dilation. Specific examples include sets that do not contain any geometric progression of length $k$ with integer ratio, sets that do not contain any geometric progression of length $k$ with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., nontrivial sets of the form $\{a, ar, as, ars\}$.

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