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

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 $\mathcal{F}_1, \ldots, \mathcal{F}_N$, each consisting of $n$ mutually disjoint segments in $\mathcal{J}_U$, there is a set $\{A_1, \ldots, A_n\}$ of $n$ disjoint segments in $\bigcup_{1\leq i\leq N}\mathcal{F}_i$ and distinct integers $p_1, \ldots, p_n\in \{1, \ldots, N\}$ satisfying that $A_j\in \mathcal{F}_{p_j}$ for all $j\in \{1, \ldots, n\}$. We generalize this property for underlying lines on fixed $k$ directions to $k$ families of simple curves with certain conditions.

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