On crossing-families in planar point sets

A $k$-crossing family in a point set $S$ general position is of $k$ segments spanned by points such that all mutually cross. In this short note we present two statements on crossing families which are based sets small cardinality: (1) Any at least 15 contains size 4. (2) There $n$ do not contain larger than $8\lceil \frac{n}{41} \rceil$. Both results improve the previously best known bounds.