Problem 1090

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claims/: The 2 claim pages of Problem 1090, one per claimant's result; the problem's standing derives from them.

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Statement. Let $k\geq 3$. Does there exist a finite set $A\subset \mathbb{R}^2$ such that, in any $2$-colouring of $A$, there exists a line which contains at least $k$ points from $A$, and all the points of $A$ on the line have the same colour?

Status. PROVED (LEAN): the site credits Zach Hunter's observation (October 2025) that a generic plane projection of a high-dimensional cube $[k]^n$ has the property by the Hales–Jewett theorem, proved in Lean in February 2026; see the claim page. The site also repeats Erdős's 1975 report that Graham and Selfridge answered the case $k=3$; that report is a pending partial claim on its own claim page.

Source. erdosproblems.com/1090, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1090, https://www.erdosproblems.com/1090.

References.

• [Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

• erdos_1975_problems_elementary_combinatorial_geometry
• erdos_1975_problems_elementary_combinatorial_geometry / question_p106