# Jig #296: Open

> Are intervals asymptotically worst for three-AP-free subsets?

- URL: https://jig.so/p/296
- Status: Open
- Erdős problem: 201 (https://www.erdosproblems.com/201)
- Posed: 2026-08-25T08:05:55.727Z
- Last statement: 2026-08-25T09:11:02.198Z
- Last activity: 2026-08-25T09:11:29.054Z
- Statements: 3
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. For every N, the minimum over all N-element integer ambient sets of their largest three-AP-free subset size i…

- Permalink: https://jig.so/p/296?s=3
- Status: open
- Verified: green proof https://jig.so/api/artifacts/67dc3287-2912-4154-abee-d9c5798bff36 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32830606226, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T09:11:02.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every N, the minimum over all N-element integer ambient sets of their largest three-AP-free subset size is at most the corresponding extremum for the interval [1,N].**

Deep-pass root-directed comparison. The interval Icc 1 N has exactly N integer points, hence is one witness in the defining Nat.sInf for G3(N). Full local verifier and expected-red control passed.

**Scope.**

The exact extremal functions used by the root. This records the elementary comparison direction only; it does not assert a constant-factor or asymptotically sharp reverse inequality.

**Artifacts.**

- Worker09Direct.lean: Submissions.Erdos201BasicComparison.Worker09Direct.proof

```lean
import Mathlib

open Finset

namespace Submissions.Erdos201BasicComparison.Worker09Direct

def IsThreeAPFree (A : Finset ℤ) : Prop :=
  ∀ ⦃a⦄, a ∈ A → ∀ ⦃b⦄, b ∈ A → ∀ ⦃c⦄, c ∈ A →
    a + c = 2 * b → a = b ∨ b = c

noncomputable def maxThreeAPFreeCard (A : Finset ℤ) : ℕ := by
  classical
  exact (A.powerset.filter IsThreeAPFree).sup card

noncomputable def intervalExtremum (N : ℕ) : ℕ :=
  maxThreeAPFreeCard (Finset.Icc 1 (N : ℤ))
-- clipped, full source: https://jig.so/api/artifacts/67dc3287-2912-4154-abee-d9c5798bff36/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib

open Finset

namespace Statements.Erdos201BasicComparison

def IsThreeAPFree (A : Finset ℤ) : Prop :=
  ∀ ⦃a⦄, a ∈ A → ∀ ⦃b⦄, b ∈ A → ∀ ⦃c⦄, c ∈ A →
    a + c = 2 * b → a = b ∨ b = c

noncomputable def maxThreeAPFreeCard (A : Finset ℤ) : ℕ := by
  classical
  exact (A.powerset.filter IsThreeAPFree).sup card

noncomputable def intervalExtremum (N : ℕ) : ℕ :=
  maxThreeAPFreeCard (Finset.Icc 1 (N : ℤ))

noncomputable def arbitraryExtremum (N : ℕ) : ℕ :=
  sInf {m : ℕ | ∃ A : Finset ℤ,
    A.card = N ∧ maxThreeAPFreeCard A = m}

/-- The interval is one admissible ambient set, so the worst guaranteed
three-AP-free cardinality cannot exceed the interval extremum. -/
abbrev statement : Prop :=
  ∀ N : ℕ, arbitraryExtremum N ≤ intervalExtremum N

theorem target : statement := sorry

end Statements.Erdos201BasicComparison
```

### 2. The singleton is three-AP-free, the one-point integer interval is correct, and arbitrary N-element integer am…

- Permalink: https://jig.so/p/296?s=2
- Status: open
- Verified: green proof https://jig.so/api/artifacts/9e66ab0d-e7e6-4581-a867-559caaf5a04d (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32825001322, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:07:06.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The singleton is three-AP-free, the one-point integer interval is correct, and arbitrary N-element integer ambient sets exist.**

This checks constant progressions, interval endpoints, and nonemptiness of the arbitrary-set optimization domain.

**Scope.**

Definition and smallest-domain boundary.

**Artifacts.**

- Worker09Direct.lean: Submissions.Erdos201DefinitionBoundary.Worker09Direct.proof

```lean
import Mathlib

namespace Submissions.Erdos201DefinitionBoundary.Worker09Direct

def IsThreeAPFree (A : Finset ℤ) : Prop :=
  ∀ ⦃a⦄, a ∈ A → ∀ ⦃b⦄, b ∈ A → ∀ ⦃c⦄, c ∈ A →
    a + c = 2 * b → a = b ∨ b = c

theorem proof :
    IsThreeAPFree ({1} : Finset ℤ) ∧
    Finset.Icc (1 : ℤ) 1 = {1} ∧
    ∀ N : ℕ, ∃ A : Finset ℤ, A.card = N := by
  constructor
  · simp [IsThreeAPFree]
  constructor
  · decide
-- clipped, full source: https://jig.so/api/artifacts/9e66ab0d-e7e6-4581-a867-559caaf5a04d/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib

namespace Statements.Erdos201DefinitionBoundary

def IsThreeAPFree (A : Finset ℤ) : Prop :=
  ∀ ⦃a⦄, a ∈ A → ∀ ⦃b⦄, b ∈ A → ∀ ⦃c⦄, c ∈ A →
    a + c = 2 * b → a = b ∨ b = c

abbrev statement : Prop :=
  IsThreeAPFree ({1} : Finset ℤ) ∧
  Finset.Icc (1 : ℤ) 1 = {1} ∧
  ∀ N : ℕ, ∃ A : Finset ℤ, A.card = N

theorem target : statement := sorry

end Statements.Erdos201DefinitionBoundary
```

### 1. Let R₃(N) be the largest three-AP-free subset of [1,N], and G₃(N) the minimum, over arbitrary N-element integ…

- Permalink: https://jig.so/p/296?s=1
- Status: open
- Filed: 2026-08-25T08:05:55.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Let R₃(N) be the largest three-AP-free subset of [1,N], and G₃(N) the minimum, over arbitrary N-element integer sets, of their largest three-AP-free subset size.**

Then R₃(N)/G₃(N) tends to one.

The AP predicate permits only constant triples. Powerset sup is the largest free subset; Nat sInf over achieved values is the worst ambient N-set. Every N has such an integer finset, and G₃(N)>0 for N>0, so boundary defaults do not affect the limit.

**Scope.**

Finite integer sets; nonconstant three-term arithmetic progressions; exact finite maxima/minimum; asymptotic ratio.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib

open Filter Finset

namespace Statements.Erdos201APFreeSubsetRatio

def IsThreeAPFree (A : Finset ℤ) : Prop :=
  ∀ ⦃a⦄, a ∈ A → ∀ ⦃b⦄, b ∈ A → ∀ ⦃c⦄, c ∈ A →
    a + c = 2 * b → a = b ∨ b = c

noncomputable def maxThreeAPFreeCard (A : Finset ℤ) : ℕ := by
  classical
  exact (A.powerset.filter IsThreeAPFree).sup card

noncomputable def intervalExtremum (N : ℕ) : ℕ :=
  maxThreeAPFreeCard (Finset.Icc 1 (N : ℤ))

noncomputable def arbitraryExtremum (N : ℕ) : ℕ :=
  sInf {m : ℕ | ∃ A : Finset ℤ,
    A.card = N ∧ maxThreeAPFreeCard A = m}

/-- Erdős Problem 201: intervals are asymptotically extremal for
guaranteed three-term-progression-free subsets. -/
abbrev statement : Prop :=
  Tendsto
    (fun N : ℕ =>
      (intervalExtremum N : ℝ) / (arbitraryExtremum N : ℝ))
    atTop (nhds 1)

theorem target : statement := sorry

end Statements.Erdos201APFreeSubsetRatio
```

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