FormalConjectures.ErdosProblems.«100»

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import FormalConjecturesUtil

Erdős Problem 100

References:

erdosproblems.com/100

[Kanold](No references found)

[GuKa15](Guth, Larry and Katz, Nets Hawk, On the Erd\H{o}s distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.)

[Piepmeyer](No references found)


open Set Metric Filter Real

open scoped EuclideanGeometry


namespace Erdos100

If two distances in A differ, they differ by at least 1.

def DistancesSeparated (A : Finset ℝ²) : Prop :=
  ∀ p₁ q₁ p₂ q₂, p₁ ∈ A → q₁ ∈ A → p₂ ∈ A → q₂ ∈ A →
    dist p₁ q₁ ≠ dist p₂ q₂ →
    |dist p₁ q₁ - dist p₂ q₂| ≥ 1

Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?

@[category research open, AMS 52]
theorem declaration uses 'sorry'erdos_100 :
    answer(sorry) ↔ ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
      A.card = n →
      DistancesSeparated A →
      diam (A : Set ℝ²) > C * n := by⊢ True ↔ ∃ C > 0, ∀ᶠ (n : ℕ) in atTop, ∀ (A : Finset ℝ²), A.card = n → DistancesSeparated A → diam ↑A > C * ↑n
  sorryAll goals completed! 🐙

Stronger conjecture: diameter $\geq n - 1$ for sufficiently large $n$.

@[category research open, AMS 52]
theorem declaration uses 'sorry'erdos_100.variants.strong :
    ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
      A.card = n →
      DistancesSeparated A →
      diam (A : Set ℝ²) ≥ n - 1 := by⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (A : Finset ℝ²), A.card = n → DistancesSeparated A → diam ↑A ≥ ↑n - 1
  sorryAll goals completed! 🐙

From [Kanold]: diameter $\geq n^{3/4}$. TODO: find reference

@[category research solved, AMS 52]
theorem declaration uses 'sorry'erdos_100.variants.kanold :
    ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
      A.card = n →
      DistancesSeparated A →
      diam (A : Set ℝ²) ≥ (n : ℝ) ^ (3 / 4 : ℝ) := by⊢ ∃ C > 0, ∀ᶠ (n : ℕ) in atTop, ∀ (A : Finset ℝ²), A.card = n → DistancesSeparated A → diam ↑A ≥ ↑n ^ (3 / 4)
  sorryAll goals completed! 🐙

From [GuKa15]: diameter $\gg n / \log n$.

@[category research solved, AMS 52]
theorem declaration uses 'sorry'erdos_100.variants.guth_katz :
    ∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∀ A : Finset ℝ²,
      A.card = n →
      DistancesSeparated A →
      diam (A : Set ℝ²) ≥ C * n / log n := by⊢ ∃ C > 0, ∀ᶠ (n : ℕ) in atTop, ∀ (A : Finset ℝ²), A.card = n → DistancesSeparated A → diam ↑A ≥ C * ↑n / log ↑n
  sorryAll goals completed! 🐙

From [Piepmeyer]: 9 points with diameter $< 5$. TODO: find reference

@[category research solved, AMS 52, formal_proof using formal_conjectures at "https://github.com/theaustinhatfield/formal-conjectures/blob/solve-erdos-100-piepmeyer/FormalConjectures/ErdosProblems/100.lean"]
theorem declaration uses 'sorry'erdos_100_piepmeyer :
    ∃ A : Finset ℝ², A.card = 9 ∧ DistancesSeparated A ∧
      diam (A : Set ℝ²) < 5 := by⊢ ∃ A, A.card = 9 ∧ DistancesSeparated A ∧ diam ↑A < 5
  sorryAll goals completed! 🐙


end Erdos100