On Erdős–Szekeres-Type Problems for k-convex Point Sets

Abstract

We study Erdős–Szekeres-type problems for k-convex point sets, a recently introduced notion that naturally extends the concept of convex position. A finite set S of n points is k-convex if there exists a spanning simple polygonization of S such that the intersection of any straight line with its interior consists of at most k connected components. We address several open problems about k-convex point sets. In particular, we extend the well-known Erdős–Szekeres Theorem by showing that, for every fixed \(k \in \mathbb {N}\), every set of n points in the plane in general position (with no three collinear points) contains a k-convex subset of size at least \(\varOmega (\log ^k{n})\). We also show that there are arbitrarily large 3-convex sets of n points in the plane in general position whose largest 1-convex subset has size \(O(\log {n})\). This gives a solution to a problem posed by Aichholzer et al. [2].