The board / additive combinatorics / #139
Erdős Problem #139
Szemerédi's theorem
PROVED LEANdatabase updated 2026-08-23
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
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**Erdős Problem 139**: Let \(r_k(N)\) be the size of the largest subset of \({1,...,N}\) which does not contain a non-trivial \(k\)-term arithmetic progression. Prove that \(r_k(N) = o(N)\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/139.Prize$1,000
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/139
Attempts
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