1. FormalConjectures
  2. ErdosProblems
  3. «105»
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Erdős Problem 105

References:

    erdosproblems.com/105

    [Be83] Beck, József, On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry. Combinatorica (1983), 281-297.

    [ErPu95] Erdős, P. and Purdy, G., Two combinatorial problems in the plane. Discrete Comput. Geom. (1995), 441-443.

    [SzTr83] Szemerédi, Endre and Trotter, Jr., William T., Extremal problems in discrete geometry. Combinatorica (1983), 381-392.

open EuclideanGeometrynamespace Erdos105

Let $A,B\subset \mathbb{R}^2$ be disjoint sets of size $n$ and $n-3$ respectively, with not all of $A$ contained on a single line. Is there a line which contains at least two points from $A$ and no points from $B$?

This has been disproved by Xichuan in the comments, who has found three explicit counterexamples.

@[category research solved, AMS 5 52, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos105.lean"] theorem erdos_105 : answer(False) ↔ ∀ A B : Finset ℝ², Disjoint A B → A.card = B.card + 3 → ¬ Collinear ℝ (A : Set ℝ²) → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] := ⊢ False ↔ ∀ (A B : Finset ℝ²), Disjoint A B → A.card = B.card + 3 → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] All goals completed! 🐙

A construction of Hickerson shows that this fails with $n-2$.

@[category research solved, AMS 5 52] theorem erdos_105.variants.hickerson : ¬ ∀ A B : Finset ℝ², Disjoint A B → A.card = B.card + 2 → ¬ Collinear ℝ (A : Set ℝ²) → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] := ⊢ ¬∀ (A B : Finset ℝ²), Disjoint A B → A.card = B.card + 2 → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] All goals completed! 🐙

A result independently proved by Beck [Be83] and Szemerédi and Trotter [SzTr83] (see [211]) implies it is true with $n-3$ replaced by $cn$ for some constant $c>0$.

@[category research solved, AMS 5 52] theorem erdos_105.variants.beck_szemeredi_trotter : ∃ c > (0 : ℝ), ∀ A B : Finset ℝ², Disjoint A B → (B.card : ℝ) ≤ c * A.card → ¬ Collinear ℝ (A : Set ℝ²) → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] := ⊢ ∃ c > 0, ∀ (A B : Finset ℝ²), Disjoint A B → ↑B.card ≤ c * ↑A.card → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] All goals completed! 🐙

It remains possible that this holds with $n-4$ (or in general with $n-O(1)$ or $(1-o(1))n$).

@[category research open, AMS 5 52] theorem erdos_105.variants.sub_four : answer(sorry) ↔ ∀ A B : Finset ℝ², Disjoint A B → A.card = B.card + 4 → ¬ Collinear ℝ (A : Set ℝ²) → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] := ⊢ True ↔ ∀ (A B : Finset ℝ²), Disjoint A B → A.card = B.card + 4 → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ line[ℝ, p, q] All goals completed! 🐙end Erdos105