The paper demonstrates that every set of n points in the plane contains a crossing family of size approximately n/79.

A new paper on the #arXiv: together with Joachim Orthaber we give the first explicit bound for linear crossing families by showing that every set of n points in the plane contains a crossing family of size roughly n/79. We also give other bounds on the sizes of crossing families depending on some parameters like the crossing number. https://arxiv.org/abs/2610.38658

Larger Crossing Families https://arxiv.org/abs/2610.38658v1 Authors: Joachim Orthaber, Patrick SchniderWe give the first explicit constant for crossing families of linear size by showing that every set of $n$ points in general position in the plane contains a crossing family of size approximately $frac{n}{79}$. We further identify some conditions where there are even larger crossing families. The first such condition forbids the existence of a large diffe