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We present some problems and results about variants of sunflowers in families of sets. In particular, we improve an upper bound of the first author, Körner and Monti on the maximum number of binary vectors of length n so that every four of them are split into two pairs by some coordinate. We also propose a weaker version of the Erdős-Rado sunflower conjecture.
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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.
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- PAR ID:
- 10273894
- Editor(s):
- Buchin, Kevin and
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 189
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 37:1--37:13
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.SoCG.2021.37
