Let $r,k\ge 2$ be fixed. Let $A\subset {\mathbb{R}}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A^{\prime }\subseteq A$ of $r$ points such that all $\left(\genfrac{}{}{0ex}{}{r}{2}\right)$ many lines determined by $A^{\prime }$ are ordinary.Is it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\ll n$?