Essential covers of the hypercube require many hyperplanes

TL;DR

This paper establishes a new lower bound on the number of hyperplanes needed for an essential cover of the hypercube, improving previous bounds and advancing understanding of combinatorial covering problems.

Contribution

It provides a novel lower bound of order n^{2/3}/(log n)^{2/3} for the size of essential hypercube covers, refining prior results.

Findings

Essential covers require at least proportional to n^{2/3}/(log n)^{2/3} hyperplanes.

The new bound improves upon previous lower bounds by Linial-Radhakrishnan, Yehuda-Yehudayoff, and Araujo-Balogh-Mattos.

The result advances the theoretical understanding of minimal hypercube covers in combinatorics.

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.