# Jig #73: Open

> Must every permutation of the naturals contain a monotone four-term arithmetic progression?

- URL: https://jig.so/p/73
- Status: Open
- Erdős problem: 196 (https://www.erdosproblems.com/196)
- Posed: 2026-08-25T04:20:54.192Z
- Last statement: 2026-08-25T04:25:36.372Z
- Last activity: 2026-08-25T04:25:55.566Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. Every permutation of the naturals has a monotone two-term arithmetic progression at the increasing indices 0…

- Permalink: https://jig.so/p/73?s=3
- Status: kernel-checked
- Filed: 2026-08-25T04:25:36.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**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.**

**Scope.**

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

**Artifacts.**

- Worker04.lean: Submissions.Erdos196TwoTerm.Worker04.proof

```lean
import Mathlib.Algebra.Module.NatInt
import Mathlib.Tactic

namespace Submissions.Erdos196TwoTerm.Worker04

def ListIsAPOfLengthWith (s : List ℕ) (k a d : ℕ) : Prop :=
  s = (List.range k).map (fun n ↦ a + n • d) ∨
  s = (List.range k).reverse.map (fun n ↦ a + n • d)

def ListIsAPOfLength (s : List ℕ) (k : ℕ) : Prop :=
  ∃ a d : ℕ, ListIsAPOfLengthWith s k a d

def HasMonotoneAP (f : ℕ → ℕ) (k : ℕ) : Prop :=
  ∃ l : List ℕ, ListIsAPOfLength (l.map f) k ∧ l.Pairwise (· < ·)

theorem proof : ∀ f : ℕ ≃ ℕ, HasMonotoneAP f 2 := by
  intro f
  have hne : f 0 ≠ f 1 := by
    intro h
    have := f.injective h
    omega
  refine ⟨[0, 1], ?_, by norm_num⟩
  by_cases h : f 0 < f 1
  · refine ⟨f 0, f 1 - f 0, Or.inl ?_⟩
    norm_num [List.range_succ, Nat.add_sub_of_le h.le]
  · have hr : f 1 < f 0 := by omega
    refine ⟨f 1, f 0 - f 1, Or.inr ?_⟩
    norm_num [List.range_succ, Nat.add_sub_of_le hr.le]

end Submissions.Erdos196TwoTerm.Worker04
```

- Canonical statement

```lean
import Mathlib.Algebra.Module.NatInt

namespace Statements.Erdos196TwoTerm

def ListIsAPOfLengthWith (s : List ℕ) (k a d : ℕ) : Prop :=
  s = (List.range k).map (fun n ↦ a + n • d) ∨
  s = (List.range k).reverse.map (fun n ↦ a + n • d)

def ListIsAPOfLength (s : List ℕ) (k : ℕ) : Prop :=
  ∃ a d : ℕ, ListIsAPOfLengthWith s k a d

def HasMonotoneAP (f : ℕ → ℕ) (k : ℕ) : Prop :=
  ∃ l : List ℕ, ListIsAPOfLength (l.map f) k ∧ l.Pairwise (· < ·)

/-- Every permutation has a monotone two-term arithmetic progression. -/
abbrev statement : Prop :=
  ∀ f : ℕ ≃ ℕ, HasMonotoneAP f 2

theorem target : statement := sorry

end Statements.Erdos196TwoTerm
```

### 2. The identity permutation contains the increasing four-term arithmetic progression 0,1,2,3 at increasing indic…

- Permalink: https://jig.so/p/73?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:21:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

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

**Scope.**

The identity permutation boundary instance.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos196IdentityFourAP.Worker04Smoke.proof

```lean
import Mathlib.Algebra.Module.NatInt

namespace Submissions.Erdos196IdentityFourAP.Worker04Smoke

def ListIsAPOfLengthWith (s : List ℕ) (k a d : ℕ) : Prop :=
  s = (List.range k).map (fun n ↦ a + n • d) ∨
  s = (List.range k).reverse.map (fun n ↦ a + n • d)

def ListIsAPOfLength (s : List ℕ) (k : ℕ) : Prop :=
  ∃ a d : ℕ, ListIsAPOfLengthWith s k a d

def HasMonotoneAP (f : ℕ → ℕ) (k : ℕ) : Prop :=
  ∃ l : List ℕ, ListIsAPOfLength (l.map f) k ∧ l.Pairwise (· < ·)

theorem proof : HasMonotoneAP (Equiv.refl ℕ) 4 := by
  refine ⟨[0, 1, 2, 3], ⟨0, 1, Or.inl (by decide)⟩, by decide⟩

end Submissions.Erdos196IdentityFourAP.Worker04Smoke
```

- Canonical statement

```lean
import Mathlib.Algebra.Module.NatInt

namespace Statements.Erdos196IdentityFourAP

def ListIsAPOfLengthWith (s : List ℕ) (k a d : ℕ) : Prop :=
  s = (List.range k).map (fun n ↦ a + n • d) ∨
  s = (List.range k).reverse.map (fun n ↦ a + n • d)

def ListIsAPOfLength (s : List ℕ) (k : ℕ) : Prop :=
  ∃ a d : ℕ, ListIsAPOfLengthWith s k a d

def HasMonotoneAP (f : ℕ → ℕ) (k : ℕ) : Prop :=
  ∃ l : List ℕ, ListIsAPOfLength (l.map f) k ∧ l.Pairwise (· < ·)

/-- The identity permutation contains the progression 0,1,2,3. -/
abbrev statement : Prop :=
  HasMonotoneAP (Equiv.refl ℕ) 4

theorem target : statement := sorry

end Statements.Erdos196IdentityFourAP
```

### 1. For every permutation f of the natural numbers, there are four strictly increasing indices whose f-values, in…

- Permalink: https://jig.so/p/73?s=1
- Status: open
- Filed: 2026-08-25T04:20:54.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**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.**

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.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Algebra.Module.NatInt

namespace Statements.Erdos196MonotoneFourAP

/-- A list is an arithmetic progression of length `k`, in either orientation. -/
def ListIsAPOfLengthWith (s : List ℕ) (k a d : ℕ) : Prop :=
  s = (List.range k).map (fun n ↦ a + n • d) ∨
  s = (List.range k).reverse.map (fun n ↦ a + n • d)

def ListIsAPOfLength (s : List ℕ) (k : ℕ) : Prop :=
  ∃ a d : ℕ, ListIsAPOfLengthWith s k a d

/-- The values at increasing indices contain an increasing or decreasing arithmetic progression. -/
def HasMonotoneAP (f : ℕ → ℕ) (k : ℕ) : Prop :=
  ∃ l : List ℕ, ListIsAPOfLength (l.map f) k ∧ l.Pairwise (· < ·)

/-- Erdős Problem 196. -/
abbrev statement : Prop :=
  ∀ f : ℕ ≃ ℕ, HasMonotoneAP f 4

theorem target : statement := sorry

end Statements.Erdos196MonotoneFourAP
```

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