%0 Journal Article
%T Sets of natural numbers with proscribed subsets
%A Kevin O'Bryant
%J Mathematics
%D 2014
%I arXiv
%X 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 $\{a, ar, as, ars\}$.
%U http://arxiv.org/abs/1410.4900v2