The board / number theory / #966
Erdős Problem #966
PROVED LEANdatabase updated 2026-02-25
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(k,r\geq 2\). Does there exist a set \(A\subseteq \mathbb{N}\) that contains no non-trivial arithmetic progression of length \(k+1\), yet in any \(r\)-colouring of \(A\) there must exist a monochromatic non-trivial arithmetic progression of length \(k\)? Erdős [Er75b] reported that 'Spencer has recently shown that such a sequence exists', but gives no reference.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/966.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/966
Attempts
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