FormalConjectures.ErdosProblems.«224»

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import FormalConjecturesUtil

Erdős Problem 224

References:

erdosproblems.com/224

[DaGr62] Danzer, L. and Gr"{u}nbaum, B., "{U}ber zwei Probleme bez"{u}glich konvexer K"{o}rper von P. Erd\H{o}s und von V. L. Klee. Math. Z. (1962), 95-99.


namespace Erdos224


open scoped EuclideanGeometry

The angle $yxz$ is obtuse.

def ObtuseAt {d : ℕ} (x y z : ℝ^d) : Prop :=
  inner ℝ (y - x) (z - x) < 0

If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.

The general case was proved by Danzer and Gr"{u}nbaum [DaGr62].

@[category research solved, AMS 52,
  formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos224.lean"]
theorem declaration uses 'sorry'erdos_224 {d : ℕ} (A : Finset (ℝ^d)) (hcard : A.card = 2 ^ d + 1) :
    ∃ x y z : ℝ^d, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧
      x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ ObtuseAt x y z := byd:ℕA:Finset (ℝ^d)hcard:A.card = 2 ^ d + 1⊢ ∃ x y z, x ∈ A ∧ y ∈ A ∧ z ∈ A ∧ x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ ObtuseAt x y z
  sorryAll goals completed! 🐙


end Erdos224