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

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.