New Lower Bounds For Essential Covers Of The Cube

TL;DR

This paper improves the lower bounds on the number of hyperplanes needed for essential covers of the n-cube, advancing understanding of minimal covering configurations in high-dimensional combinatorics.

Contribution

It establishes a new lower bound of (n^{5/9}/(log n)^{4/9}) hyperplanes, refining previous bounds using advanced combinatorial methods.

Findings

Established a new lower bound for essential covers.

Improved understanding of minimal hyperplane covers.

Built on and extended previous lower bound techniques.

Abstract

An essential cover of the vertices of the $n$-cube $\{0,1\}^n$ by hyperplanes is a minimal covering where no hyperplane is redundant and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a construction of an essential cover with $\lceil \frac{n}{2} \rceil + 1$ hyperplanes and showed that $\Omega(\sqrt{n})$ hyperplanes are required. Recently, Yehuda and Yehudayoff improved the lower bound by showing that any essential cover of the $n$-cube contains at least $\Omega(n^{0.52})$ hyperplanes. In this paper, building on the method of Yehuda and Yehudayoff, we prove that $\Omega \left( \frac{n^{5/9}}{(\log n)^{4/9}} \right)$ hyperplanes are needed.