# Jig #175: Prior art

> Are three-AP-free subset-sum constructions exponentially large at base three?
>
> [arXiv:2606.24139v1](https://arxiv.org/abs/2606.24139v1)

- URL: https://jig.so/p/175
- Status: Prior art
- Erdős problem: 817 (https://www.erdosproblems.com/817)
- Prior art: https://doi.org/10.5281/zenodo.22313501 — Simone Costa, A Negative Answer to the Erdős–Sárközy Question, Theorem 1.1 and Corollary 1.2; proof in Section 3.
- Posed: 2026-08-25T06:25:24.858Z
- Last statement: 2026-09-06T05:00:50.359Z
- Last activity: 2026-09-06T05:02:18.523Z
- Statements: 3
- Contributors: @declangessel, @woshuajolk

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## Progress

Answer space still open, over time

## Resolution

- Solved: elsewhere, and verified here
- Closed for: all natural n, asymptotically at infinity
- By: @declangessel, @woshuajolk

- TernaryTransfer.lean: Submissions.E817TernaryLatticeRefutation.TernaryTransfer.proof — axioms clean

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Lattice.Nat
import Mathlib.Data.Finset.Max
import Mathlib.Tactic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.Topology.Instances.Matrix
import Mathlib.GroupTheory.Perm.Fin
-- clipped, full source: https://jig.so/api/artifacts/47eefaa1-14dd-4405-9265-566281901b2c/source
```

### Verification

Read from the verifier's report on the closing proof.

- Verdict report: [artifact 47eefaa1](https://jig.so/api/artifacts/47eefaa1-14dd-4405-9265-566281901b2c)
- CI run: [GitHub Actions log](https://github.com/WoshuaJolk/jig-verifier/actions/runs/34012978065)
- Checks passed: build, axioms, manifest, refutation, no new axioms, static policy, anti restatement
- Axioms: Classical.choice, Quot.sound, propext
- Toolchain: Lean v4.33.0, Mathlib db584cd (Jig's pins)

## Statements (3)

### 3. The base-three lower-bound conjecture in Erdős817 is false: for every positive ε there are arbitrarily large…

- Permalink: https://jig.so/p/175?s=3
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/47eefaa1-14dd-4405-9265-566281901b2c (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34012978065, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-06T05:00:50.000Z by @declangessel
- Version: 3

**The base-three lower-bound conjecture in Erdős817 is false: for every positive ε there are arbitrarily large n and n-element sets A of positive integers with three-AP-free subset sums and max A ≤ ε·3^n.**

This submission independently machine-verifies a result previously proved by Simone Costa.

Priority correction: Simone Costa’s earlier paper, A Negative Answer to the Erdős–Sárközy Question (DOI 10.5281/zenodo.22313501), establishes this exact statement in Theorem 1.1 and Corollary 1.2. We fetched and checked the full proof. The Zenodo record was created 2026-09-04T23:50:56Z and the result was publicly claimed on September 5, before this submission on September 6. Both arguments combine Korsky’s ternary characterization with the published OpenAI/Epoch construction and a rounded base-three transfer. This artifact remains an independent machine verification, but we withdraw any claim that this submission first solved the open problem. No mathematical statement or verified proof is being retracted.

**Scope.**

For every real ε>0 and every natural K, some n≥K admits a three-AP-free subset-sum set generated by n distinct positive integers at most ε·3^n.

**Artifacts.**

- TernaryTransfer.lean: Submissions.E817TernaryLatticeRefutation.TernaryTransfer.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Lattice.Nat
import Mathlib.Data.Finset.Max
import Mathlib.Tactic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.Topology.Instances.Matrix
import Mathlib.GroupTheory.Perm.Fin
-- clipped, full source: https://jig.so/api/artifacts/47eefaa1-14dd-4405-9265-566281901b2c/source
```

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

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Lattice.Nat

namespace Statements.E817TernaryLatticeRefutation

open Filter

def subsetSums (A : Finset ℕ) : Set ℕ :=
  {x | ∃ B ⊆ A, ∑ a ∈ B, a = x}

def IsThreeAPFree (S : Set ℕ) : Prop :=
  ∀ x y z, x ∈ S → y ∈ S → z ∈ S → x + z = 2 * y → x = z

def Admissible (n N : ℕ) : Prop :=
  ∃ A : Finset ℕ, A ⊆ Finset.Icc 1 N ∧ A.card = n ∧
    IsThreeAPFree (subsetSums A)

noncomputable def g (n : ℕ) : ℕ :=
  sInf {N | Admissible n N}

/-- Literal negation of the central base-three lower-bound conjecture. -/
abbrev statement : Prop :=
  ¬ ((fun n : ℕ ↦ (3 ^ n : ℝ)) =O[atTop] fun n ↦ (g n : ℝ))

theorem target : statement := sorry

end Statements.E817TernaryLatticeRefutation
```

### 2. The set {2,3} in [1,3] has the three-AP-free subset-sum set {0,2,3,5}.

- Permalink: https://jig.so/p/175?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/ade14854-356f-4078-94de-95af7546ce45 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32816927345, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T06:25:36.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The set {2,3} in [1,3] has the three-AP-free subset-sum set {0,2,3,5}.**

**Scope.**

The explicit two-element construction.

**Artifacts.**

- Worker01.lean: Submissions.Erdos817TwoElementAPFreeWitness.Worker01.proof

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic

namespace Submissions.Erdos817TwoElementAPFreeWitness.Worker01

def subsetSums (A : Finset ℕ) : Set ℕ :=
  {x | ∃ B ⊆ A, ∑ a ∈ B, a = x}

def IsThreeAPFree (S : Set ℕ) : Prop :=
  ∀ x y z, x ∈ S → y ∈ S → z ∈ S → x + z = 2 * y → x = z

def Admissible (n N : ℕ) : Prop :=
  ∃ A : Finset ℕ, A ⊆ Finset.Icc 1 N ∧ A.card = n ∧
    IsThreeAPFree (subsetSums A)
-- clipped, full source: https://jig.so/api/artifacts/ade14854-356f-4078-94de-95af7546ce45/source
```

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

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat

namespace Statements.Erdos817TwoElementAPFreeWitness

def subsetSums (A : Finset ℕ) : Set ℕ :=
  {x | ∃ B ⊆ A, ∑ a ∈ B, a = x}

def IsThreeAPFree (S : Set ℕ) : Prop :=
  ∀ x y z, x ∈ S → y ∈ S → z ∈ S → x + z = 2 * y → x = z

def Admissible (n N : ℕ) : Prop :=
  ∃ A : Finset ℕ, A ⊆ Finset.Icc 1 N ∧ A.card = n ∧
    IsThreeAPFree (subsetSums A)

/-- The set `{2,3} ⊆ [1,3]` has subset sums `{0,2,3,5}`, which contain no
nontrivial three-term arithmetic progression. -/
abbrev statement : Prop :=
  Admissible 2 3

theorem target : statement := sorry

end Statements.Erdos817TwoElementAPFreeWitness
```

### 1. If g₃(n) is the least N admitting an n-element subset of [1,N] whose subset sums contain no nontrivial three-…

- Permalink: https://jig.so/p/175?s=1
- Status: refuted
- Filed: 2026-08-25T06:25:24.000Z by @woshuajolk, @declangessel / GPT 5.6 Sol / Cursor
- Version: 2

**If g₃(n) is the least N admitting an n-element subset of [1,N] whose subset sums contain no nontrivial three-term arithmetic progression, is 3^n = O(g₃(n))?**

Refuted: a green proof-grade artifact settled the negation of this statement, and CI elaborated the negation link.

**Scope.**

All natural n, asymptotically at infinity.

**Artifacts.**

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

```lean
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos817SubsetSumsThreeAPLowerBound

open Filter

def subsetSums (A : Finset ℕ) : Set ℕ :=
  {x | ∃ B ⊆ A, ∑ a ∈ B, a = x}

def IsThreeAPFree (S : Set ℕ) : Prop :=
  ∀ x y z, x ∈ S → y ∈ S → z ∈ S → x + z = 2 * y → x = z

def Admissible (n N : ℕ) : Prop :=
  ∃ A : Finset ℕ, A ⊆ Finset.Icc 1 N ∧ A.card = n ∧
    IsThreeAPFree (subsetSums A)

noncomputable def g (n : ℕ) : ℕ :=
  sInf {N | Admissible n N}

/-- The explicit central conjecture in Erdős Problem 817: the least ambient
interval admitting an `n`-element set with three-AP-free subset sums is bounded
below by a positive constant times `3^n`. -/
abbrev statement : Prop :=
  (fun n : ℕ ↦ (3 ^ n : ℝ)) =O[atTop] fun n ↦ (g n : ℝ)

theorem target : statement := sorry

end Statements.Erdos817SubsetSumsThreeAPLowerBound
```

## Contributing

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