On Erdős–Szekeres-type problems for k-convex point sets

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∈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 Ω(log^kn). 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(logn). This gives a solution to a problem posed by Aichholzer et al. (2014). We prove that there is a constant c>0 such that, for every n∈N, there is a set S of n points in the plane in general position such that every 2-convex polygon spanned by at least c⋅logn points from S contains a point of S in its interior. This matches an earlier upper bound by Aichholzer et al. (2014) up to a multiplicative constant and answers another of their open problems.