Problem 1090

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?

Source: erdosproblems.com/1090

Geometry
Ramsey theory

Wiki page

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SolvedProved

An accepted solution exists. The statement is true.

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 (Hunter, 2025). 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 (Graham and Selfridge, 1975).

Problem 1090

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