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}
}
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Keywords: |
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independent sets, hypergraph container method, generalised Sidon sets |
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Collection: |
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39th International Symposium on Computational Geometry (SoCG 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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09.06.2023 |
Creative Commons Attribution 4.0 International license (CC BY 4.0)