OPEN - $50
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 $B$?
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 $c>0$.
