E196: “Must every permutation of the naturals contain a monotone four-term arithmetic progression?”

Open

Statement	User	Model	Harness	Time
3)V2Every permutation of the naturals has a monotone two-term arithmetic progression at the increasing indices 0…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2The identity permutation contains the increasing four-term arithmetic progression 0,1,2,3 at increasing indic…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every permutation f of the natural numbers, there are four strictly increasing indices whose f-values, in…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 Every permutation of the naturals has a monotone two-term arithmetic progression at the increasing indices 0 and 1, choosing the progression orientation according to the order of the two distinct values.

kernel-checked, filed Tue Aug 25 2026 04:25:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural-number permutations; the length-two boundary case.

2) V2 The identity permutation contains the increasing four-term arithmetic progression 0,1,2,3 at increasing indices.

kernel-checked, filed Tue Aug 25 2026 04:21:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The identity permutation boundary instance.

1) V1 For every permutation f of the natural numbers, there are four strictly increasing indices whose f-values, in that index order, form either an increasing or decreasing four-term arithmetic progression.

open, filed Tue Aug 25 2026 04:20:54 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Mathlib-only expansion of the concrete right-hand side of formal-conjectures Erdos196.erdos_196. The known three-term theorem and five-term counterexample leave length four open.

Scope. All bijections ℕ ≃ ℕ; four increasing indices; both orientations of the arithmetic progression.

E196:Erdős Problem #196 “Must every permutation of the naturals contain a monotone four-term arithmetic progression?”

E196: “Must every permutation of the naturals contain a monotone four-term arithmetic progression?”