Proved: a conjecture of Erdős and Graham. They observed the lower bound $f(n)\ge(\frac12+o(1))\frac n{\log n}$ (for every $m$, which one may take below $\binom{n+1}2$, the multiples of the least prime not dividing $m$, a prime below $(2+o(1))\log n$, avoid $m$ as a subset sum), and Alon and Freiman [AlFr88] proved the matching upper bound by exhibiting an $m$, the least common multiple of the integers below $s$ with $s$ largest such that $m\le n^2/(20\log^2n)$, whose avoiding sets have at most $(\frac12+o(1))\frac n{\log n}$ elements. The site labels the problem PROVED and credits the paper. Claim page: Alon and Freiman 1988 (accepted, refereed in Combinatorica).