Some Results Related to a Conjecture of Dirac's

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of n pseudolines has no member incident to more than 4n/9 points of intersection. (This shows the “Strong Dirac” conjecture to be false for pseudolines.) We also prove non-trivial lower bounds on the maximum number of intersection points on any curve in an arrangement of curves in the plane, for various classes of curves. (This shows that analogs to the “Weak Dirac” theorem apply for these classes of curves.)