FormalConjectures.ErdosProblems.«1071»

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import FormalConjecturesUtil

Erdős Problem 1071

References:

erdosproblems.com/1071

[Da85] Danzer, L., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985), 167-177.


open Set Metric EuclideanGeometry Order


namespace Erdos1071

Two segments are disjoint if they only intersect at their endpoints (if at all).

def SegmentsDisjoint (seg1 seg2 : ℝ² × ℝ²) : Prop :=
  segment ℝ seg1.1 seg1.2 ∩ segment ℝ seg2.1 seg2.2 ⊆ {seg1.1, seg1.2, seg2.1, seg2.2}

Can a finite set of disjoint unit segments in a unit square be maximal? Solved affirmatively by [Da85], who gave an explicit construction.

This was formalized in Lean by Alexeev using Aristotle and ChatGPT.

@[category research solved, AMS 52, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1071.lean"]
theorem declaration uses 'sorry'erdos_1071.parts.i :
    answer(True) ↔ ∃ S : Finset (ℝ² × ℝ²),
      Maximal (fun T : Finset (ℝ² × ℝ²) =>
        (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧
          seg.1 0 ∈ Icc 0 1 ∧ seg.1 1 ∈ Icc 0 1 ∧
          seg.2 0 ∈ Icc 0 1 ∧ seg.2 1 ∈ Icc 0 1) ∧
          (T : Set (ℝ² × ℝ²)).Pairwise SegmentsDisjoint) S := by⊢ True ↔
  ∃ S,
    Maximal
      (fun T =>
        (∀ seg ∈ T,
            dist seg.1 seg.2 = 1 ∧
              seg.1.ofLp 0 ∈ Icc 0 1 ∧ seg.1.ofLp 1 ∈ Icc 0 1 ∧ seg.2.ofLp 0 ∈ Icc 0 1 ∧ seg.2.ofLp 1 ∈ Icc 0 1) ∧
          (↑T).Pairwise SegmentsDisjoint)
      S
  sorryAll goals completed! 🐙

Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite? Solved affirmatively by [Fo99], who gave an explicit construction.

This was formalized in Lean by Alexeev using Aristotle and ChatGPT.

@[category research solved, AMS 52, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos1071b.lean"]
theorem declaration uses 'sorry'erdos_1071.parts.ii :
    answer(sorry) ↔ ∃ (R : Set ℝ²) (S : Set (ℝ² × ℝ²)),
      IsOpen R ∧ IsConnected R ∧ S.Countable ∧ S.Infinite ∧
      Maximal (fun T : Set (ℝ² × ℝ²) =>
        (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ seg.1 ∈ R ∧ seg.2 ∈ R) ∧
        T.Pairwise SegmentsDisjoint) S := by⊢ True ↔
  ∃ R S,
    IsOpen R ∧
      IsConnected R ∧
        S.Countable ∧
          S.Infinite ∧
            Maximal (fun T => (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ seg.1 ∈ R ∧ seg.2 ∈ R) ∧ T.Pairwise SegmentsDisjoint) S
  sorryAll goals completed! 🐙


end Erdos1071