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On symmetric intersecting families of vectors

Part of: Extremal combinatorics Algebraic combinatorics

Published online by Cambridge University Press:  18 March 2021

Sean Eberhard ,
Jeff Kahn ,
Bhargav Narayanan  and
Sophie Spirkl
Sean Eberhard
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: eberhard.math@gmail.com)
Jeff Kahn
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA (e-mails: jkahn@math.rutgers.edu, sophie.spirkl@rutgers.edu)
Bhargav Narayanan*
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA (e-mails: jkahn@math.rutgers.edu, sophie.spirkl@rutgers.edu)
Sophie Spirkl
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA (e-mails: jkahn@math.rutgers.edu, sophie.spirkl@rutgers.edu)
*
*Corresponding author. Email: narayanan@math.rutgers.edu
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Abstract

A family of vectors in [k]n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k]n invariant under a transitive group of symmetries is o(kn), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 05E18: Group actions on combinatorial structures
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 6 , November 2021 , pp. 899 - 904
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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