Weak Dirac conjecture on ordinary lines

Let $\mathcal{P}$ be a set of $n$ non-collinear points in the plane, and let $\mathcal{L}(\mathcal{P})$ be the set of lines determined by pairs of points of $\mathcal{P}$. A point is incident to the lines of $\mathcal{L}(\mathcal{P})$ that pass through it.

Weak Dirac conjecture. Every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least

$$\left\lceil\frac n2\right\rceil$$

lines of $\mathcal{L}(\mathcal{P})$.

The conjecture is used in the paper to obtain a lower bound for the distinct-angle quantity $A(n)$. It remains open, although significant progress is known.