Erdős-Szekeres-type theorems for segments and noncrossing convex sets
A family \(\mathcal F\) of convex sets is in convex position if none of its members is contained in the convex hull of the others. In this paper, the authors prove that there exists an integer \(N=N(n)>0\) such that every family of at least \(N\) pairwise noncrossing convex bodies in the plane of which any three are in convex position, has \(n\) members in convex position. This statement cannot be generalized to families of convex bodies whose boundaries may have four intersection points per pair. It is shown by the authors that there is an infinite family of straight-line segments in the plane such that any three of them are in convex position but no four are. On the other hand, it is also proved that for every \(n\), there exists an integer \(M=M(n)>0\) with the following property. Every family of at least \(M\) straight-line segments in the plane such that any four of them are in convex position, has \(n\) members in convex position.
scientific article; zbMATH DE number 1505826