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On symmetric 3-wise intersecting families


Authors: David Ellis and Bhargav Narayanan
Journal: Proc. Amer. Math. Soc. 145 (2017), 2843-2847
MSC (2010): Primary 05D05; Secondary 05E18
DOI: https://doi.org/10.1090/proc/13452
Published electronically: January 23, 2017
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Abstract: A family of sets is said to be symmetric if its automorphism group is transitive, and $ 3$-wise intersecting if any three sets in the family have nonempty intersection. Frankl conjectured in 1981 that if $ \mathcal {A}$ is a symmetric $ 3$-wise intersecting family of subsets of $ \{1,2,\dots ,n\}$, then $ \vert\mathcal {A}\vert = o(2^n)$. Here, we give a short proof of Frankl's conjecture using a `sharp threshold' result of Friedgut and Kalai.


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Additional Information

David Ellis
Affiliation: School of Mathematical Sciences, Queen Mary, University of London, London E14NS, United Kingdom
Email: d.ellis@qmul.ac.uk

Bhargav Narayanan
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB30WB, United Kingdom
Email: b.p.narayanan@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/proc/13452
Received by editor(s): August 18, 2016
Published electronically: January 23, 2017
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2017 American Mathematical Society