A published paper or arXiv preprint answers the main question of Erdős Problem #131 (theorem erdos_131 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

Let F(N) be the maximal size of A \subseteq \{1, \dots, N\} such that no a \in A divides the sum of any distinct elements of A \setminus \{a\}. Is it true that F(N) > N^{1/2 - o(1)}? The answer is no, by the bound of Pham and Zakharov [PhZa24] (`erdos_131.variants.pham_zakharov`).

References formal-conjectures lists for this problem (a starting point, not evidence):

• [ELRSS99] Erdős, P. and Lev, V. and Rauzy, G. and Sándor, C. and Sárközy, A., Greedy algorithm, arithmetic progressions, subset sums and divisibility. Discrete Math. (1999), 119-135.
• [PhZa24] Pham, H. T. and Zakharov, D., Sharp bound for the Erdős–Straus non-averaging set problem. arXiv:2410.14624 (2024).