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Essential covers of the hypercube require many hyperplanes

Part of: Extremal combinatorics

Published online by Cambridge University Press:  16 December 2024

Lisa Sauermann  and
Zixuan Xu
Lisa Sauermann
Affiliation:
University of Bonn, Bonn, Germany
Zixuan Xu*
Affiliation:
Massachusetts Institute of Technology, Cambridge, USA
*
Corresponding author: Zixuan Xu; Email: zixuanxu@mit.edu
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Abstract

We prove a new lower bound for the almost 20-year-old problem of determining the smallest possible size of an essential cover of the $n$-dimensional hypercube $\{\pm 1\}^n$, that is, the smallest possible size of a collection of hyperplanes that forms a minimal cover of $\{\pm 1\}^n$ and such that, furthermore, every variable appears with a non-zero coefficient in at least one of the hyperplane equations. We show that such an essential cover must consist of at least $10^{-2}\cdot n^{2/3}/(\log n)^{2/3}$ hyperplanes, improving previous lower bounds of Linial–Radhakrishnan, of Yehuda–Yehudayoff, and of Araujo–Balogh–Mattos.

Keywords

Essential covershypercube

MSC classification

Primary: 05D40: Probabilistic methods

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 34 , Issue 3 , May 2025 , pp. 326 - 337
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

*

Research supported in part by NSF Award DMS-2100157 and a Sloan Research Fellowship.

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