Erdős Problem #12 — Reciprocal sums when no term divides a sum of two larger terms

Erdős and Sárközy asked whether such a set can have size about the square root of $N$, and whether every such set must be much smaller than $N$ infinitely often. The problem page records affirmative and negative answers claimed through a DeepMind construction, while still labeling the problem open.

Known results

• Erdős and Sárközy, 1970: every such infinite set has density $0$.
• Erdős and Sárközy, 1970: examples have $|A\cap[1,N]|>N/f(N)$ infinitely often for every $f(N)\to\infty$.
• Elsholtz and Planitzer, 2017: $|A\cap[1,N]|\gg N^{1/2}/((\log N)^{1/2}(\log\log N)^2(\log\log\log N)^2)$.
• Schoen, 2001, and Baier, 2004: pairwise-coprime sets satisfy upper bounds $\ll N^{2/3}$ and $\ll N^{2/3}/\log N$ infinitely often.

April 2026 DeepMind construction

The construction is reported to give $|A\cap[1,N]|\ge N/(\log N)^{O(\log\log\log N)}$ for all sufficiently large $N$. Thus the first question is answered yes and the second no. The associated Lean declarations remain by sorry, so this is an unverified claim rather than a checkable resolution.

Current status (as of April 2026): The reported construction settles both growth questions if validated, but no independently checkable proof was found; the reciprocal-sum question remains open.