Problem 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 $B$?
Categories:
Geometry
Progress
Conjectured by Erdős and Purdy [ErPu95] (the prize is for a proof or disproof). A construction of Hickerson shows that this fails with $n-2$. 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 $c>0$.This has been disproved by Xichuan in the comments, who has found three explicit counterexamples. It remains possible that this holds with $n-4$ (or in general with $n-O(1)$ or $(1-o(1))n$).
Source: erdosproblems.com/105 | Last verified: January 13, 2026