Erdős Problem #1090 — Monochromatic Collinear Sets in Two-Colorings

The problem asks whether, for every $k \ge 3$, some finite planar point set forces every two-coloring to contain a monochromatic line with at least $k$ points. No proposer or date is identified in the available sources.

Known results

• Gruslys showed that the analogous statement fails for three colors, and therefore for every number of colors at least $3$.
• His construction gives, for every $n \ge 2$, an $n$-point set with no four collinear points that admits a coloring with no monochromatic line.
• The case $k=2$ follows from the Motzkin–Rabin theorem; the problem concerns $k \ge 3$.
• The three-color construction does not settle the two-color case.

Current status (as of September 2026): The two-color problem remains open; no proof or counterexample for the stated formulation has been found.