On a problem of Erd\H{o}s and S\'ark\"ozy about sequences with no term dividing the sum of two larger terms

TL;DR

This paper proves a longstanding conjecture by Erdős and Sárközy that sequences with property P have size at most roughly one-third of the set size, confirming the upper bound for large n.

Contribution

The authors resolve the open problem by proving the exact upper bound on the size of sequences with property P for sufficiently large n.

Findings

Confirmed the conjectured upper bound for large n

Established the exact maximum size of sequences with property P

Resolved a problem posed by Erdős and Sárközy in 1970

Abstract

In 1970, Erd\H{o}s and S\'ark\"ozy wrote a joint paper studying sequences of integers $a_1<a_2<\dots$ having what they called property P, meaning that no $a_i$ divides the sum of two larger $a_j,a_k$. In the paper, it was stated that the authors believed, but could not prove, that a subset $A\subset[n]$ with property P has cardinality at most $|A|\leqslant \left\lfloor \frac{n}{3}\right\rfloor+1$. In 1997, Erd\H{o}s offered \$100 for a proof or disproof of the claim that $|A|\leqslant \frac{n}{3}+C$, for some absolute constant $C$. We resolve this problem, and in fact prove that $|A|\leqslant\left\lfloor \frac{n}{3}\right\rfloor+1$ for $n$ sufficiently large.