License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.59
URN: urn:nbn:de:0030-drops-179097
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17909/
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Suk, Andrew ; Zeng, Ji

On Higher Dimensional Point Sets in General Position

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LIPIcs-SoCG-2023-59.pdf (0.6 MB)

Abstract

A finite point set in ℝ^d is in general position if no d + 1 points lie on a common hyperplane. Let α_d(N) be the largest integer such that any set of N points in ℝ^d with no d + 2 members on a common hyperplane, contains a subset of size α_d(N) in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that α₂(N) < N^{5/6 + o(1)}. In this paper, we also use the container method to obtain new upper bounds for α_d(N) when d ≥ 3. More precisely, we show that if d is odd, then α_d(N) < N^{1/2 + 1/(2d) + o(1)}, and if d is even, we have α_d(N) < N^{1/2 + 1/(d-1) + o(1)}.
We also study the classical problem of determining the maximum number a(d,k,n) of points selected from the grid [n]^d such that no k + 2 members lie on a k-flat. For fixed d and k, we show that a(d,k,n)≤ O(n^{d/{2⌊(k+2)/4⌋}(1- 1/{2⌊(k+2)/4⌋d+1})}), which improves the previously best known bound of O(n^{d/⌊(k + 2)/2⌋}) due to Lefmann when k+2 is congruent to 0 or 1 mod 4.

BibTeX - Entry

@InProceedings{suk_et_al:LIPIcs.SoCG.2023.59,
  author =	{Suk, Andrew and Zeng, Ji},
  title =	{{On Higher Dimensional Point Sets in General Position}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{59:1--59:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17909},
  URN =		{urn:nbn:de:0030-drops-179097},
  doi =		{10.4230/LIPIcs.SoCG.2023.59},
  annote =	{Keywords: independent sets, hypergraph container method, generalised Sidon sets}
}

Keywords: independent sets, hypergraph container method, generalised Sidon sets
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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