Sunflowers in Set Systems of Bounded Dimension
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.
Sunflower
VC-dimension
Littlestone dimension
pseudodisks
Mathematics of computing~Combinatoric problems
37:1-37:13
Regular Paper
We would like to thank Amir Yehudayoff for suggesting working with the Littlestone dimension, and the SoCG 2021 referees for helpful comments.
Jacob
Fox
Jacob Fox
Department of Mathematics, Stanford University, CA, USA
Supported by a Packard Fellowship and by NSF award DMS-1855635.
János
Pach
János Pach
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Moscow Institute of Physics and Technology, Moscow, Russia
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."
Andrew
Suk
Andrew Suk
Department of Mathematics, University of California San Diego, La Jolla, CA, USA
Supported by NSF CAREER award DMS-1800746, NSF award DMS-1952786, and an Alfred Sloan Fellowship.
10.4230/LIPIcs.SoCG.2021.37
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Jacob Fox, János Pach, and Andrew Suk
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