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OP-8883OpenMathematics · general

Erdős #195 · arithmetic progressions

What is the largest kk such that in any permutation of Z\mathbb{Z} there must exist a monotone kk-term arithmetic progression x1<⋯<xkx_1<\cdots<x_k?

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

What is the largest kk such that in any permutation of Z\mathbb{Z} there must exist a monotone kk-term arithmetic progression x1<⋯<xkx_1<\cdots<x_k?

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Background

Geneson [Ge19] proved that k≤5k\leq 5. Adenwalla [Ad22] proved that k≤4k\leq 4.

See also [194] and [196].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #195 link
  2. [2]Upstream Lean 4 statement for Erdős #195 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 195 link

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