Erdős–Szekeres-Type Theorems for Segments and Noncrossing Convex Sets

AbstractA family $$\mathcal{F}$$ of convex sets is said to be in convex position, if none of its members is contained in the convex hull of the others. It is proved that there is a function N(n) with the following property. If $$\mathcal{F}$$ is a family of at least N(n) plane convex sets with nonempty interiors, such that any two members of $$\mathcal{F}$$ have at most two boundary points in common and any three are in convex position, then $$\mathcal{F}$$ has n members in convex position. This result generalizes a theorem of T. Bisztriczky and G. Fejes Tóth. The statement does not remain true, if two members of $$\mathcal{F}$$ may share four boundary points. This follows from the fact that there exist infinitely many straight-line segments such that any three are in convex position, but no four are. However, there is a function M(n) such that every family of at least M(n) segments, any four of which are in convex position, has n members in convex position.