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 nonzero vectors in the plane and JU is the set of all line segments in direction u for some u∈U, then for every N families F1,…,FN, each consisting of n mutually disjoint segments in JU, there is a set {A1,…,An} of n disjoint segments in ⋃1≤i≤NFi and distinct integers p1,…,pn∈{1,…,N} satisfying that Aj∈Fpj for all j∈{1,…,n}. We generalize this property for underlying lines on fixed k directions to k families of simple curves with certain conditions.