The board / additive combinatorics / #817
Erdős Problem #817
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Define \(g_k(n)\) to be the minimal \(N\) such that \(\{1, ..., N\}\) contains some \(A\) of size \(|A| = n\) such that \[ \langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\} \] contains no non-trivial \(k\)-term arithmetic progression.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/817.Prizenone
Topicsadditive combinatorics
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/817
Attempts
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Live page status is OPEN, but “Currently working on this problem” lists SamKorsky (accessed 2026-07-28); stopping to avoid colliding with a current worker.
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