The independence and clique cover numbers of the squarefree graph

TL;DR

This paper determines the maximum size of a subset of integers where no pair's product is squarefree, resolving a longstanding problem posed by Erdős and Sárközy in 1992.

Contribution

It precisely characterizes the largest subset avoiding squarefree products, specifically identifying the complement of odd squarefree numbers as optimal.

Findings

Maximum subset size is achieved by the complement of odd squarefree numbers.

Resolved a problem posed by Erdős and Sárközy in 1992.

Provides a complete characterization of the extremal set.

Abstract

We determine the largest subset $A\subseteq \{1,\dotsc,n\}$ such that for all $a,b\in A$, the product $ab$ is not squarefree. Specifically, the maximum size is achieved by the complement of the odd squarefree numbers. This resolves a problem of Paul Erd\H{o}s and Andr\'as S\'ark\"ozy from 1992.