Abstract
Given a family \({\mathcal {F}}\) of k-element sets, \(S_1,\ldots ,S_r\in {\mathcal {F}}\) form an r-sunflower if \(S_i \cap S_j =S_{i'} \cap S_{j'}\) for all \(i \ne j\) and \(i' \ne j'\). According to a famous conjecture of Erdős and Rado (JAMA 35: 85–90, 1960), there is a constant \(c=c(r)\) such that if \(|{\mathcal {F}}|\ge c^k\), then \({\mathcal {F}}\) contains an r-sunflower. We come close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension, \(\text {VC-dim}({\mathcal {F}})\le d\). In this case, we show that r-sunflowers exist under the slightly stronger assumption \(|{\mathcal {F}}|\ge 2^{10k(dr)^{2\log ^{*} k}}\). Here, \(\log ^*\) denotes the iterated logarithm function. We also verify the Erdős-Rado conjecture for families \({\mathcal {F}}\) of bounded Littlestone dimension and for some geometrically defined set systems.
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Acknowledgements
We would like to thank Amir Yehudayoff for suggesting working with the Littlestone dimension, and the SoCG 2021 referees for helpful comments.
Funding
Fox: Supported by a Packard Fellowship and by NSF Awards DMS-1800053 and DMS-2154169. Pach: Supported by NKFIH Grant Nos. K-131529, KKP-133864, Austrian Science Fund Z 342-N31, Ministry of Education and Science of the Russian Federation Mega Grant No. 075-15-2019-1926, ERC Advanced Grant “GeoScape.” Suk: Supported an NSF CAREER award, NSF award DMS-1952786, and an Alfred Sloan Fellowship.
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Fox, J., Pach, J. & Suk, A. Sunflowers in Set Systems of Bounded Dimension. Combinatorica 43, 187–202 (2023). https://doi.org/10.1007/s00493-023-00012-z
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DOI: https://doi.org/10.1007/s00493-023-00012-z