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On Erdős-Szekeres-type problems for \(k\)-convex point sets. (English) Zbl 1448.52013

Summary: 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 \(\Omega (\log^k n)\). 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 O. Aichholzer et al. [Comput. Geom. 47, No. 8, 809–832 (2014; Zbl 1292.52001)]. 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. [loc. cit.] up to a multiplicative constant and answers another of their open problems.

MSC:

52C10 Erdős problems and related topics of discrete geometry

Citations:

Zbl 1292.52001
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References:

[1] Aichholzer, Oswin; Aurenhammer, Franz; Demaine, Erik D.; Hurtado, Ferran; Ramos, Pedro; Urrutia, Jorge, On \(k\)-convex polygons, Comput. Geom., 45, 3, 73-87 (2012) · Zbl 1244.52005
[2] Aichholzer, Oswin; Aurenhammer, Franz; Hackl, Thomas; Hurtado, Ferran; Pilz, Alexander; Ramos, Pedro; Urrutia, Jorge; Valtr, Pavel; Vogtenhuber, Birgit, On \(k\)-convex point sets, Comput. Geom., 47, 8, 809-832 (2014) · Zbl 1292.52001
[3] Aichholzer, Oswin; Balko, Martin; Hackl, Thomas; Pilz, Alexander; Ramos, Pedro; Valtr, Pavel; Vogtenhuber, Birgit, Holes in 2-convex point sets, Comput. Geom., 74, 38-49 (2018) · Zbl 1432.52030
[4] Bárány, Imre; Valtr, Pavel, A positive fraction Erdős-Szekeres theorem, Discrete Comput. Geom., 19, 3, 335-342 (1998), (special issue) · Zbl 0914.52007
[5] Erdős, Paul, Some more problems on elementary geometry, Austral. Math. Soc. Gaz., 5, 2, 52-54 (1978) · Zbl 0417.52002
[6] Erdős, Paul; Szekeres, George, A combinatorial problem in geometry, Compos. Math., 2, 463-470 (1935) · Zbl 0012.27010
[7] Erdős, Paul; Szekeres, George, On some extremum problems in elementary geometry, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 3-4, 53-62 (1960) · Zbl 0103.15502
[8] Gerken, Tobias, Empty convex hexagons in planar point sets, Discrete Comput. Geom., 39, 1-3, 239-272 (2008) · Zbl 1184.52016
[9] Harborth, Heiko, Konvexe Fünfecke in ebenen Punktmengen, Elem. Math., 33, 5, 116-118 (1978) · Zbl 0397.52005
[10] Holmsen, Andreas F.; Mojarrad, Hossein Nassajian; Pach, János; Tardos, Gábor, Two extensions of the Erdős-Szekeres problem (2017), Preliminary version: http://arxiv.org/abs/1710.11415
[11] Horton, Joseph D., Sets with no empty convex \(7\)-gons, Canad. Math. Bull., 26, 4, 482-484 (1983) · Zbl 0521.52010
[12] Matoušek, Jiří, (Lectures on Discrete Geometry. Lectures on Discrete Geometry, Graduate Texts in Mathematics, vol. 212 (2002), Springer-Verlag: Springer-Verlag New York), xvi+481 · Zbl 0999.52006
[13] Nicolás, Carlos M., The empty hexagon theorem, Discrete Comput. Geom., 38, 2, 389-397 (2007) · Zbl 1146.52010
[14] Pór, Attila; Valtr, Pavel, The partitioned version of the Erdős-Szekeres theorem, Discrete Comput. Geom., 28, 4, 625-637 (2002) · Zbl 1019.52011
[15] Suk, Andrew, On the Erdős-Szekeres convex polygon problem, J. Amer. Math. Soc., 30, 4, 1047-1053 (2017) · Zbl 1370.52032
[16] Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M., Elementary Real Analysis (2001), Prentice-Hall · Zbl 0872.26001
[17] Valtr, Pavel, Convex independent sets and \(7\)-holes in restricted planar point sets, Discrete Comput. Geom., 7, 2, 135-152 (1992) · Zbl 0748.52005
[18] Valtr, Pavel, Sets in \(\mathbf{R}^d\) with no large empty convex subsets, Discrete Math., 108, 1-3, 115-124 (1992) · Zbl 0766.52003
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