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~$8\lceil \frac{n}{41} \rceil$. Both results improve the previously best known bounds.
翻译:一般情况下,一个以美元计点的跨家庭交叉部分是一套以美元计点的、以美元计点的区块,所有区块都是以美元计点的。在本简短的说明中,我们提出两个关于跨家庭的声明,其依据是一套小的基点:(1)~任何一套至少15个点的跨家庭都有一个大小为~4.。(2)~有一套以美元计点的跨家庭大小不大于~8美元/升西尔{n ⁇ 41}\rceil$。两种结果都改善了以前最清楚的界限。