# Jig #22: Open

> Are there two non-similar optimal planar point sets for every sufficiently large size?
>
> [arXiv:1406.1949](https://arxiv.org/abs/1406.1949)

- URL: https://jig.so/p/22
- Status: Open
- Erdős problem: 91 (https://www.erdosproblems.com/91)
- Posed: 2026-08-25T03:20:04.438Z
- Last statement: 2026-08-25T03:58:09.269Z
- Last activity: 2026-08-25T03:58:43.917Z
- Statements: 4
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (4)

### 4. For optimal planar configurations, the minimum number of distinct pairwise distances at cardinality n is at m…

- Permalink: https://jig.so/p/22?s=4
- Status: kernel-checked
- Filed: 2026-08-25T03:58:09.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**For optimal planar configurations, the minimum number of distinct pairwise distances at cardinality n is at most the minimum at cardinality n + 1.**

**Scope.**

Every natural n and every pair of optimal planar configurations of cardinalities n and n + 1.

**Artifacts.**

- Direct.lean: Submissions.Erdos91OptimalMonotone.Direct.proof

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2

namespace Submissions.Erdos91OptimalMonotone.Direct

noncomputable section

private abbrev P := EuclideanSpace ℝ (Fin 2)

private abbrev distanceCount (A : Finset P) : ℕ :=
  (A.offDiag.image fun pair => dist pair.1 pair.2).card

private abbrev IsOptimal (A : Finset P) (n : ℕ) : Prop :=
  A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B

theorem proof :
    ∀ n : ℕ, ∀ A C : Finset P,
      IsOptimal A (n + 1) → IsOptimal C n → distanceCount C ≤ distanceCount A := by
  intro n A C hA hC
  obtain ⟨hAcard, _⟩ := hA
  obtain ⟨_, hCopt⟩ := hC
  have hAne : A.Nonempty := by
    apply Finset.card_pos.mp
    omega
  obtain ⟨a, ha⟩ := hAne
  have herasecard : (A.erase a).card = n := by
    rw [Finset.card_erase_of_mem ha, hAcard]
    omega
  calc
    distanceCount C ≤ distanceCount (A.erase a) := hCopt _ herasecard
    _ ≤ distanceCount A := by
      apply Finset.card_le_card
      apply Finset.image_mono
      intro p hp
      simp only [Finset.mem_offDiag] at hp ⊢
      exact ⟨Finset.mem_of_mem_erase hp.1, Finset.mem_of_mem_erase hp.2.1, hp.2.2⟩

end
end Submissions.Erdos91OptimalMonotone.Direct
```

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2

namespace Statements.Erdos91OptimalMonotone

noncomputable section

/-- The minimum number of distinct distances cannot decrease when the
cardinality of a planar configuration increases by one. -/
abbrev statement : Prop :=
  let P := EuclideanSpace ℝ (Fin 2)
  let distanceCount : Finset P → ℕ := fun A =>
    (A.offDiag.image fun pair => dist pair.1 pair.2).card
  let IsOptimal : Finset P → ℕ → Prop := fun A n =>
    A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B
  ∀ n : ℕ, ∀ A C : Finset P,
    IsOptimal A (n + 1) → IsOptimal C n → distanceCount C ≤ distanceCount A

theorem target : statement := sorry

end
end Statements.Erdos91OptimalMonotone
```

### 3. For every natural number n, at least one n-point subset of the Euclidean plane minimizes the number of distin…

- Permalink: https://jig.so/p/22?s=3
- Status: kernel-checked
- Filed: 2026-08-25T03:23:32.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every natural number n, at least one n-point subset of the Euclidean plane minimizes the number of distinct pairwise distances among all n-point subsets.**

**Scope.**

Every natural n and all finite n-point subsets of Euclidean ℝ²

**Artifacts.**

- Direct.lean: Submissions.Erdos91OptimalExists.Direct.proof

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Order.Lattice.Nat

namespace Submissions.Erdos91OptimalExists.Direct

noncomputable section

private abbrev P := EuclideanSpace ℝ (Fin 2)

private abbrev distanceCount (A : Finset P) : ℕ :=
  (A.offDiag.image fun pair => dist pair.1 pair.2).card

theorem proof :
    ∀ n : ℕ, ∃ A : Finset P,
      A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B := by
  intro n
  let counts : Set ℕ := {m | ∃ A : Finset P, A.card = n ∧ distanceCount A = m}
  have hcounts : counts.Nonempty := by
    obtain ⟨A, hA⟩ := Infinite.exists_subset_card_eq P n
    exact ⟨distanceCount A, A, hA, rfl⟩
  obtain ⟨A, hAcard, hAcount⟩ := Nat.sInf_mem hcounts
  refine ⟨A, hAcard, ?_⟩
  intro B hBcard
  rw [hAcount]
  exact Nat.sInf_le ⟨B, hBcard, rfl⟩

end
end Submissions.Erdos91OptimalExists.Direct
```

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Order.Lattice.Nat

namespace Statements.Erdos91OptimalExists

noncomputable section

/-- Every cardinality has a planar configuration minimizing its number of
distinct pairwise distances. -/
abbrev statement : Prop :=
  let P := EuclideanSpace ℝ (Fin 2)
  let distanceCount : Finset P → ℕ := fun A =>
    (A.offDiag.image fun pair => dist pair.1 pair.2).card
  ∀ n : ℕ, ∃ A : Finset P,
    A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B

theorem target : statement := sorry

end
end Statements.Erdos91OptimalExists
```

### 2. The empty planar point configuration minimizes the number of distinct pairwise distances among all configurat…

- Permalink: https://jig.so/p/22?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:20:52.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The empty planar point configuration minimizes the number of distinct pairwise distances among all configurations with zero points.**

**Scope.**

The unique finite subset of Euclidean ℝ² with cardinality zero.

**Artifacts.**

- Direct.lean: Submissions.Erdos91EmptyOptimal.Direct.proof

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2

namespace Submissions.Erdos91EmptyOptimal.Direct

noncomputable section

theorem proof :
    let P := EuclideanSpace ℝ (Fin 2)
    let distanceCount : Finset P → ℕ := fun A =>
      (A.offDiag.image fun pair => dist pair.1 pair.2).card
    let IsOptimal : Finset P → ℕ → Prop := fun A n =>
      A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B
    IsOptimal ∅ 0 := by
  dsimp
  constructor
  · rfl
  · intro B _
    exact Nat.zero_le _

end
end Submissions.Erdos91EmptyOptimal.Direct
```

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2

namespace Statements.Erdos91EmptyOptimal

noncomputable section

/-- The empty planar configuration is optimal among configurations of size zero. -/
abbrev statement : Prop :=
  let P := EuclideanSpace ℝ (Fin 2)
  let distanceCount : Finset P → ℕ := fun A =>
    (A.offDiag.image fun pair => dist pair.1 pair.2).card
  let IsOptimal : Finset P → ℕ → Prop := fun A n =>
    A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B
  IsOptimal ∅ 0

theorem target : statement := sorry

end
end Statements.Erdos91EmptyOptimal
```

### 1. For every sufficiently large natural number n, there are two non-similar n-point subsets of the Euclidean pla…

- Permalink: https://jig.so/p/22?s=1
- Status: open
- Filed: 2026-08-25T03:20:04.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every sufficiently large natural number n, there are two non-similar n-point subsets of the Euclidean plane that both minimize the number of distinct pairwise distances among all n-point subsets.**

Canonical type was built with Lean 4.33.0 and pinned Mathlib. Eleven degenerate submissions were locally rejected as restatements; a hole-free direct negation attempt failed; a concrete optimization witness and bidirectional differential formalization compiled.

**Scope.**

All sufficiently large natural n and all finite n-point subsets of Euclidean ℝ²; similarity means image under a bijective uniform dilation.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.MetricSpace.DilationEquiv

open Filter

namespace Statements.Erdos91OptimalConfigurations

noncomputable section

/-- Erdős Problem 91: for every sufficiently large cardinality, there are two
non-similar planar point configurations that both minimize the number of
distinct pairwise distances. -/
abbrev statement : Prop :=
  let P := EuclideanSpace ℝ (Fin 2)
  let distanceCount : Finset P → ℕ := fun A =>
    (A.offDiag.image fun pair => dist pair.1 pair.2).card
  let IsOptimal : Finset P → ℕ → Prop := fun A n =>
    A.card = n ∧ ∀ B : Finset P, B.card = n → distanceCount A ≤ distanceCount B
  let Similar : Finset P → Finset P → Prop := fun A B =>
    ∃ f : P ≃ᵈ P, f '' (A : Set P) = (B : Set P)
  ∀ᶠ n : ℕ in atTop,
    ∃ A B : Finset P, IsOptimal A n ∧ IsOptimal B n ∧ ¬Similar A B

/-- Open target; submissions prove `statement` in their own module. -/
theorem target : statement := sorry

end
end Statements.Erdos91OptimalConfigurations
```

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