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Sunflowers in Set Systems of Bounded Dimension

Authors Jacob Fox, János Pach, Andrew Suk

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Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, CA, USA
János Pach
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Moscow Institute of Physics and Technology, Moscow, Russia
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

  • Fox, Jacob: Supported by a Packard Fellowship and by NSF award DMS-1855635.
  • Pach, János: Rényi Institute, Budapest and MIPT, Moscow. Supported by NKFIH grants K-176529, KKP-133864, Austrian Science Fund Z 342-N31, Ministry of Education and Science of the Russian Federation MegaGrant No. 075-15-2019-1926, ERC Advanced Grant "GeoScape."
  • Suk, Andrew: Supported by NSF CAREER award DMS-1800746, NSF award DMS-1952786, and an Alfred Sloan Fellowship.

Acknowledgements

We would like to thank Amir Yehudayoff for suggesting working with the Littlestone dimension, and the SoCG 2021 referees for helpful comments.

Cite AsGet BibTex

Jacob Fox, János Pach, and Andrew Suk. Sunflowers in Set Systems of Bounded Dimension. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.37

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Sunflower
  • VC-dimension
  • Littlestone dimension
  • pseudodisks

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