The ordinary-line conjecture for configurations on many lines

Let a Sylvester–Gallai configuration be a configuration of points and lines in the complex affine plane in which every line determined by two points contains a third point. Suppose the configuration lies on $m>3$ lines, and let $C(m)$ be a function of $m$.

Ordinary-line conjecture. If each of those lines contains more than $C(m)$ points, then the set admits an ordinary line, that is, a line containing exactly two points of the set.

This conjecture generalizes the paper’s theorem for configurations on concurrent lines to arbitrary lines and is described as substantially more difficult. The lower bound on the number of points per line is needed because the Fermat configurations form an infinite family lying on only three lines. The conjecture remains open.