FormalConjectures.ErdosProblems.«99»

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import FormalConjecturesUtil

Erdős Problem 99

References:

erdosproblems.com/99

[BeFo99] Bezdek, Andr'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.

[Er94b] Erd\H{o}s, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.

open Set Metric EuclideanGeometry


namespace Erdos99

A set has minimum distance $1$ if all pairwise distances are at least $1$, and the minimum is achieved.

def HasMinDist1 (A : Finset ℝ²) : Prop :=
  (∀ᵉ (p ∈ A) (q ∈ A), p ≠ q → dist p q ≥ 1) ∧
  (∃ᵉ (p ∈ A) (q ∈ A), dist p q = 1)

Three points form an equilateral triangle of side length 1.

def FormsEquilateralTriangle (p q r : ℝ²) : Prop :=
  dist p q = 1 ∧ dist q r = 1 ∧ dist p r = 1

For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?

@[category research open, AMS 52]
theorem declaration uses 'sorry'erdos_99 :
    answer(sorry) ↔ ∀ᶠ n in Filter.atTop, ∀ A : Finset ℝ²,
      A.card = n → HasMinDist1 A →
      (IsMinOn (fun B: Finset ℝ² => diam (B : Set ℝ²)) {B : Finset ℝ² | B.card = n ∧ HasMinDist1 B} A) →
      ∃ᵉ (p ∈ A) (q ∈ A) (r ∈ A), FormsEquilateralTriangle p q r := by⊢ True ↔
  ∀ᶠ (n : ℕ) in Filter.atTop,
    ∀ (A : Finset ℝ²),
      A.card = n →
        HasMinDist1 A →
          IsMinOn (fun B => diam ↑B) {B | B.card = n ∧ HasMinDist1 B} A →
            ∃ p ∈ A, ∃ q ∈ A, ∃ r ∈ A, FormsEquilateralTriangle p q r
sorryAll goals completed! 🐙


end Erdos99