Sets of natural numbers with proscribed subsets

TL;DR

This paper investigates upper bounds on the size of subsets of natural numbers avoiding specific configurations, such as geometric progressions and multiplicative squares, providing new bounds for families closed under dilation.

Contribution

It introduces new upper bounds for the maximum size of subsets avoiding certain geometric and multiplicative patterns, especially for families closed under dilation.

Findings

Derived new upper bounds for sets avoiding geometric progressions with integer and rational ratios.

Established bounds for sets excluding multiplicative squares.

Analyzed families of subsets closed under dilation to generalize the bounds.

Abstract

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 $\{a, ar, as, ars\}$.