The board / number theory / #344
Erdős Problem #344
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
If \(A\subseteq \mathbb{N}\) is a set of integers such that \[\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1/2}\] for all \(N\) then must \(A\) be subcomplete? That is, must \[P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}\] contain an infinite arithmetic progression? This is true, and was proved by Szemerédi and Vu [SzVu06]. The implied constant is absolute, as in [SzVu06b, Corollary 1.4].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/344.Prizenone
Topicsnumber theory, complete sequences
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/344
Attempts
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