@inproceedings{995409dca7ad4a96bd56703ccf10e4bd,

title = "On Erd{\H o}s–Szekeres-Type Problems for k-convex Point Sets",

abstract = "We study Erd{\H o}s–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 Erd{\H o}s–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(logk 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.{\^A} [2]. 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 · log n points from S contains a point of S in its interior. This matches an earlier upper bound by Aichholzer et al.{\^A} [2] up{\^A} to a multiplicative constant and answers another of their open problems.",

keywords = "Convex position, Point set, k-convex point set, k-convex polygon",

author = "Martin Balko and Sujoy Bhore and {Mart{\'i}nez Sandoval}, Leonardo and Pavel Valtr",

note = "Publisher Copyright: {\textcopyright} 2019, Springer Nature Switzerland AG.; 30th International Workshop on Combinatorial Algorithms, IWOCA 2019 ; Conference date: 23-07-2019 Through 25-07-2019",

year = "2019",

month = jan,

day = "1",

doi = "10.1007/978-3-030-25005-8_4",

language = "English",

isbn = "9783030250041",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "35--47",

editor = "Colbourn, {Charles J.} and Roberto Grossi and Nadia Pisanti",

booktitle = "Combinatorial Algorithms - 30th International Workshop, IWOCA 2019, Proceedings",

address = "Germany",

}