Problem 105

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claims/: The 1 claim page of Problem 105, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $B$?

Status. DISPROVED (LEAN): the site credits three explicit counterexamples posted in its comments by Xichuan in October 2025, one of which (twelve points against nine) was later formalized in Lean in Boris Alexeev's repository; see the claim page.

Source. erdosproblems.com/105, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #105, https://www.erdosproblems.com/105.

References.

• [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.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• szemeredi_1983_extremal_problems_discrete_geometry