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.2021.37
URN: urn:nbn:de:0030-drops-138366
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13836/
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Fox, Jacob ; Pach, János ; Suk, Andrew

Sunflowers in Set Systems of Bounded Dimension

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LIPIcs-SoCG-2021-37.pdf (0.6 MB)

Abstract

Given a family F of k-element sets, S₁,…,S_r ∈ F form an r-sunflower if S_i ∩ S_j = S_{i'} ∩ S_{j'} for all i ≠ j and i' ≠ j'. According to a famous conjecture of Erdős and Rado (1960), there is a constant c = c(r) such that if |F| ≥ c^k, then F contains an r-sunflower.
We come close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension, VC-dim(F) ≤ d. In this case, we show that r-sunflowers exist under the slightly stronger assumption |F| ≥ 2^{10k(dr)^{2log^{*} k}}. Here, log^* denotes the iterated logarithm function.
We also verify the Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems.

BibTeX - Entry

@InProceedings{fox_et_al:LIPIcs.SoCG.2021.37,
  author =	{Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew},
  title =	{{Sunflowers in Set Systems of Bounded Dimension}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{37:1--37:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13836},
  URN =		{urn:nbn:de:0030-drops-138366},
  doi =		{10.4230/LIPIcs.SoCG.2021.37},
  annote =	{Keywords: Sunflower, VC-dimension, Littlestone dimension, pseudodisks}
}

Keywords: Sunflower, VC-dimension, Littlestone dimension, pseudodisks
Collection: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue Date: 2021
Date of publication: 02.06.2021

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