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].

We prove that there is a constant $c>0$ such that, for every $n \in \mathbb {N}$, there is a set S of n points in the plane in general position such that every 2-convex polygon spanned by at least $c\cdot \log {n}$ points from S contains a point of S in its interior. This matches an earlier upper bound by Aichholzer et al. [2] up to a multiplicative constant and answers another of their open problems.

The project leading to this application has received funding from European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 678765. M. Balko and P. Valtr were supported by the grant no. 18-19158S of the Czech Science Foundation (GAČR). M. Balko and L. Martínez-Sandoval were supported by the grant 1452/15 from Israel Science Foundation. M. Balko was supported by Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). S. Bhore was supported by the Austrian Science Fund (FWF) under project number P31119. L. Martínez Sandoval was supported by the grant ANR-17-CE40-0018 of the French National Research Agency ANR (project CAPPS). This research was supported by the PRIMUS/17/SCI/3 project of Charles University.