General Position for Planar Line Arrangements and HD_2(p,3)

For every $\delta > 0$ and infinitely many $n$ there is a set of $n$ lines in the plane with no intersecting quadruple such that every subset of size at least $n^{4/5+\delta}$ contains three concurrent lines. This improves the bound for a dual form of a theorem of Balogh and Solymosi, and yields an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$.