Problem 846

Erdős Problem #846: declared status 'disproved'. Formalized: yes. Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\epsilon n$ with no three on a line. Is it true that $A$ is the union of a finite number of sets where no three are on a line? Prize: no. Tags: geometry.