The Dirac--Goodman--Pollack conjecture for pseudolines

Let $\mathcal L$ be a nontrivial arrangement of $n$ pseudolines, and let $r(\mathcal L)$ be the maximum number of crossing points, or vertices of the arrangement, lying on a pseudoline in $\mathcal L$.

Dirac--Goodman--Pollack conjecture. There is a constant $c>0$ such that

$$r(\mathcal L) \geq c n.$$

This is the pseudoline-arrangement formulation of the Dirac problem and is also described in terms of allowable sequences. In the abstract, the paper states that the conjecture is confirmed with the concrete bound $c=1/845$.