On problems related to crossing families

Given a set of points in the plane, a \emph{crossing family} is a collection of segments, each joining two of the points, such that every two segments intersect internally. Aronov et al. [Combinatorica,~14(2):127-134,~1994] proved that any set of $n$ points contains a crossing family of size $\Omega(\sqrt{n})$. They also mentioned that there exist point sets whose maximum crossing family uses at most $\frac{n}{2}$ of the points. We improve the upper bound on the size of crossing families to $5\lceil \frac{n}{24} \rceil$. We also introduce a few generalizations of crossing families, and give several lower and upper bounds on our generalized notions.