Erdős problem 105 · problems.science

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 $B$ ?

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

105.lean

Retained formal statement2 of 4

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 $c>0$ .

FormalConjectures/ErdosProblems/105.leanErdos105.erdos_105.variants.beck_szemeredi_trotter3 linesExact file

1∃ c > 0,2  ∀ (A B : Finset (EuclideanSpace ℝ (Fin 2))),3    Disjoint A B → ↑B.card ≤ c * ↑A.card → ¬Collinear ℝ ↑A → ∃ p ∈ A, ∃ q ∈ A, p ≠ q ∧ ∀ b ∈ B, b ∉ affineSpan ℝ {p, q}

SolvedStatement only, no proof

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