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Essential covers of the hypercube require many hyperplanes

Lisa Sauermann, Zixuan Xu

TL;DR

This work proves a new lower bound for the size of essential covers of the $n$-dimensional hypercube by hyperplanes with full variable participation, showing any essential cover must have at least $10^{-2}\cdot n^{2/3}/(\log n)^{2/3}$ hyperplanes. The proof combines a structured coefficient-matrix decomposition with a probabilistic uncovered-vertex construction, leveraging tools such as magnitude decompositions, Bang's lemma, and Hoeffding-type concentration to force a contradiction with essential-cover minimality. The approach refines prior methods (e.g., Yehuda--Yehudayoff, Araujo--Balogh--Mattos) and yields implications for related proof-complexity lower bounds. Overall, it advances understanding of geometric covers of the hypercube and tightens the landscape of lower bounds in this domain.

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 $\{\pm 1\}^n$, i.e. 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.

Essential covers of the hypercube require many hyperplanes

TL;DR

This work proves a new lower bound for the size of essential covers of the -dimensional hypercube by hyperplanes with full variable participation, showing any essential cover must have at least hyperplanes. The proof combines a structured coefficient-matrix decomposition with a probabilistic uncovered-vertex construction, leveraging tools such as magnitude decompositions, Bang's lemma, and Hoeffding-type concentration to force a contradiction with essential-cover minimality. The approach refines prior methods (e.g., Yehuda--Yehudayoff, Araujo--Balogh--Mattos) and yields implications for related proof-complexity lower bounds. Overall, it advances understanding of geometric covers of the hypercube and tightens the landscape of lower bounds in this domain.

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 -dimensional hypercube , i.e. the smallest possible size of a collection of hyperplanes that forms a minimal cover of 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 hyperplanes, improving previous lower bounds of Linial-Radhakrishnan, of Yehuda-Yehudayoff and of Araujo-Balogh-Mattos.
Paper Structure (5 sections, 10 theorems, 26 equations, 1 figure)

This paper contains 5 sections, 10 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Outline
  4. Finding the Uncovered vertex
  5. Decomposing the Matrix

Key Result

Theorem 1.2

For $n\ge 2$, any essential cover of the $n$-dimensional hypercube $\{\pm 1\}^n$ must consist of at least $10^{-2}\cdot n^{2/3}/(\log n)^{2/3}$ hyperplanes.

Figures (1)

  • Figure 1: The decomposition of the coefficient matrix $V$ of the essential cover

Theorems & Definitions (22)

  • Definition 1.1: Essential cover
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1: Magnitude
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: LinialR05
  • proof : Proof of \ref{['prop:skew-cover']}
  • Lemma 2.5: Bang's lemma bang1951ball1991
  • ...and 12 more