eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-06-02
37:1
37:13
10.4230/LIPIcs.SoCG.2021.37
article
Sunflowers in Set Systems of Bounded Dimension
Fox, Jacob
1
Pach, János
2
3
Suk, Andrew
4
Department of Mathematics, Stanford University, CA, USA
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Moscow Institute of Physics and Technology, Moscow, Russia
Department of Mathematics, University of California San Diego, La Jolla, CA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol189-socg2021/LIPIcs.SoCG.2021.37/LIPIcs.SoCG.2021.37.pdf
Sunflower
VC-dimension
Littlestone dimension
pseudodisks