Squares in sumsets

TL;DR

This paper investigates the maximum size of square-sum-free subsets within initial segments of natural numbers, establishing that the largest such set grows roughly as n^{1/3} as n increases.

Contribution

The paper provides an asymptotic estimate for the maximum size of square-sum-free subsets, resolving a problem posed by Erdős in 1986.

Findings

Maximum size of square-sum-free sets is of order n^{1/3+o(1)}

Established asymptotic growth rate for these sets

Answer to Erdős's problem from 1986

Abstract

A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this question, we show that this maximum cardinality is of order $n^{1/3+o(1)}$.