Erdős Problem #195 — open problem in combinatorics

Erdős Problem #195

What is the largest k such that in any permutation of ℤ there must exist a monotone k-term arithmetic progression x_1 < ⋯ < x_k? Here a permutation of ℤ is a one-sided arrangement a_1, a_2, a_3, … of the integers, i.e.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-195,
  title        = {Erdős Problem #195},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-195}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

What is the largest $k$ such that in any permutation of $Z$ there must exist a monotone $k$ -term arithmetic progression $x_{1} < \dots < x_{k}$ ?

Here a permutation of $Z$ is a one-sided arrangement $a_{1}, a_{2}, a_{3}, \dots$ of the integers, i.e. a bijection $N \to Z$ , and a monotone $k$ -term arithmetic progression is a subsequence $a_{i_{1}}, \dots, a_{i_{k}}$ with $i_{1} < \dots < i_{k}$ forming an increasing or decreasing arithmetic progression.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«195». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_195 :
    answer(sorry) = sSup {k : ℕ | ∀ f : ℕ ≃ ℤ, HasMonotoneAP f k}

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/195. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/195
[Ad22] Adenwalla, S., Avoiding Monotone Arithmetic Progressions in Permutations of Integers. arXiv:2211.04451 (2022).
[Ge19] Geneson, Jesse, Forbidden arithmetic progressions in permutations of subsets of the integers. Discrete Math. (2019), 1489-1491.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.