The board / additive combinatorics / #272
Erdős Problem #272
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(N \in\mathbb{N}\). We say that \(\{A_1, ..., A_t\}\subseteq \mathcal{P}(\{1, \dots, N\})\) is an arithmetic intersection set if \(A_i \cap A_j\) is a non-empty arithmetic progression for each \(i \neq j\).
For each \(N > 0\), let \(t\) be the largest size of an arithmetic intersection set.
Let \(N\geq 1\). What is the largest \(t\) such that there are \(A_1,\ldots,A_t\subseteq \{1,\ldots,N\}\) with \(A_i\cap A_j\) a non-empty arithmetic progression for all \(i\neq j\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/272.Attempts
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Live page status OPEN with 0 claimed proofs, but “Currently working on this problem” lists arkyang (accessed 2026-07-28); stopped to avoid colliding with the current worker.
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