FormalConjectures.ErdosProblems.«798»

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import FormalConjecturesUtil

Erdős Problem 798

References:

erdosproblems.com/798

[Al91] Alon, N., Economical coverings of sets of lattice points. Geom. Funct. Anal. (1991), 224-230.

namespace Erdos798open Filter Asymptotics

The grid ${1,\ldots,n}^2$, viewed as a set of points of the plane.

def gridPoints (n : ℕ) : Set (ℝ × ℝ) :=
  {p : ℝ × ℝ | ∃ i j : ℕ, 1 ≤ i ∧ i ≤ n ∧ 1 ≤ j ∧ j ≤ n ∧ p = ((i : ℝ), (j : ℝ))}

S is a set of points of ${1,\ldots,n}^2$ such that the lines determined by the pairs of distinct points of S cover all points of ${1,\ldots,n}^2$.

def IsLineCover (n : ℕ) (S : Set (ℝ × ℝ)) : Prop :=
  S ⊆ gridPoints n ∧
    ∀ x ∈ gridPoints n, ∃ p ∈ S, ∃ q ∈ S, p ≠ q ∧ Collinear ℝ ({p, q, x} : Set (ℝ × ℝ))

The minimum number of points in ${1,\ldots,n}^2$ such that the lines determined by these points cover all points in ${1,\ldots,n}^2$.

noncomputable def t (n : ℕ) : ℕ :=
  sInf {k | ∃ S : Set (ℝ × ℝ), IsLineCover n S ∧ S.ncard = k}

Let $t(n)$ be the minimum number of points in ${1,\ldots,n}^2$ such that the $\binom{t}{2}$ lines determined by these points cover all points in ${1,\ldots,n}^2$.

Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?

A problem of Erdős and Purdy, who proved $t(n) \gg n^{2/3}$.

Resolved by Alon [Al91] who proved $t(n) \ll n^{2/3}\log n$.

@[category research solved, AMS 5 52, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos798.lean"]
theorem erdos_798 : answer(True) ↔
    (fun n : ℕ => (t n : ℝ)) =o[atTop] (fun n : ℕ => (n : ℝ)) := by⊢ True ↔ (fun n ↦ ↑(t n)) =o[atTop] fun n ↦ ↑n
  sorryAll goals completed! 🐙

A problem of Erdős and Purdy, who proved $t(n) \gg n^{2/3}$.

@[category research solved, AMS 5 52]
theorem erdos_798.variants.lower_bound :
    (fun n : ℕ => (n : ℝ) ^ (2 / 3 : ℝ)) =O[atTop] (fun n : ℕ => (t n : ℝ)) := by⊢ (fun n ↦ ↑n ^ (2 / 3)) =O[atTop] fun n ↦ ↑(t n)
  sorryAll goals completed! 🐙

Resolved by Alon [Al91] who proved $t(n) \ll n^{2/3}\log n$.

@[category research solved, AMS 5 52]
theorem erdos_798.variants.upper_bound :
    (fun n : ℕ => (t n : ℝ)) =O[atTop]
      (fun n : ℕ => (n : ℝ) ^ (2 / 3 : ℝ) * Real.log n) := by⊢ (fun n ↦ ↑(t n)) =O[atTop] fun n ↦ ↑n ^ (2 / 3) * Real.log ↑n
  sorryAll goals completed! 🐙

end Erdos798