FormalConjectures.ErdosProblems.«101»

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import FormalConjecturesUtil

Erdős Problem 101

Reference: erdosproblems.com/101




namespace Erdos101


open EuclideanGeometry Filter Asymptotics

The set of lines in $\mathbb{R}^2$ containing exactly $k$ points from a given set $S$.

private noncomputable def linesWithPointsFor (k : ℕ) (S : Set ℝ²) : Set (AffineSubspace ℝ ℝ²) :=
  let determined_lines := { affineSpan ℝ {p, q} | (p ∈ S) (q ∈ S) (_ : p ≠ q) }
  { L ∈ determined_lines | (↑L ∩ S).ncard = k }

The maximum number of lines containing exactly $4$ points among all sets $S$ of $n$ points in $\mathbb{R}^2$ satisfying the condition that no five points are collinear.

noncomputable def numLinesWithFourPointMax (n : ℕ) : ℕ :=
  sSup {((linesWithPointsFor 4 S).ncard)| (S : Set ℝ²)
    (_ : S.ncard = n) (_ : S.Finite) (_ : NonCollinearFor 5 S)}

Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.

@[category research open, AMS 52]
theorem declaration uses 'sorry'erdos_101 : (fun n => (numLinesWithFourPointMax n : ℝ)) =o[atTop] (fun n => (n : ℝ)^2) := by⊢ (fun n => ↑(numLinesWithFourPointMax n)) =o[atTop] fun n => ↑n ^ 2
  sorryAll goals completed! 🐙


-- TODO(firsching): formalize other results from the additional material

end Erdos101