A system of disjoint representatives of line segments with given $k$ directions

Abstract

We prove that for all positive integers $n$ and $k$, there exists an integer $N=N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and ${\mathcal{J}}_U$ is the set of all line segments in direction $u$ for some $u\in U$, then for every $N$ families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_N$, each consisting of $n$ mutually disjoint segments in ${\mathcal{J}}_U$, there is a set $\{A_1,\dots ,A_n\}$ of $n$ disjoint segments in ${\bigcup }_{1\le i\le N}{\mathcal{F}}_i$ and distinct integers $p_1,\dots ,p_n\in \{1,\dots ,N\}$ satisfying that $A_j\in {\mathcal{F}}_{p_j}$ for all $j\in \{1,\dots ,n\}$. We generalize this property for underlying lines on fixed $k$ directions to $k$ families of simple curves with certain conditions.