The Dirac conjecture for point-line configurations

Let $S$ be a noncollinear set of $n$ points in the plane. A line is determined by $S$ if it contains at least two points of $S$.

Dirac's conjecture. There is a constant $c>0$ such that some point of $S$ is incident to at least $c n$ lines determined by $S$.

The conjecture is a classical incidence-geometric problem and is presented here as the modified form of Dirac's original conjecture, which asked for a point incident to at least $\frac{n}{2}$ connecting lines. The original formulation has counterexamples, while the constant-factor formulation is the subject of the paper's allowable-sequence generalization.