The board / additive combinatorics / #198
Erdős Problem #198
DISPROVED LEANdatabase updated 2025-11-23
The statement as posed is false. 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 \(V\) be a vector space over the rationals and let \(k\) be a fixed positive integer. Then there is a set \(X_k \subseteq V\) such that \(X_k\) meets every infinite arithmetic progression in \(V\) but \(X_k\) intersects every \(k\)-element arithmetic progression in at most two points. At the end of [Ba75] the author claims that by "slightly modifying the method of [his proof]", one can prove this.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/198.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/198
Attempts
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