On a problem of Erdős and Sárközy about sequences with no term dividing the sum of two larger terms

In 1970, Erdős and Sárközy 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 $\mathrm{\mid }A\mathrm{\mid }⩽\lfloor \frac{n}{3}\rfloor +1$. In 1997, Erdős offered $100 for a proof or disproof of the claim that $\mathrm{\mid }A\mathrm{\mid }⩽\frac{n}{3}+C$, for some absolute constant $C$. We resolve this problem, and in fact prove that $\mathrm{\mid }A\mathrm{\mid }⩽\lfloor \frac{n}{3}\rfloor +1$ for $n$ sufficiently large.