O. Aichholzer, J. Kyncl, M. Scheucher, B. Vogtenhuber, and
P. Valtr
A k-crossing family in a point set S in general position is a set of k segments
spanned by points of S such that all k segments mutually cross. In this short
note we present two statements on crossing families which are based on sets
of small cardinality: (1) Any set of at least 15 points contains a crossing
family of size 4. (2) There are sets of n points which do not contain a
crossing family of size larger than 8⌈n41⌉. Both results improve the
previously best known bounds.