A Lower Bound for Essential Covers of the Cube

The amount of hyperplanes that are needed in order to cover the Boolean cube has been studied in various contexts. Linial and Radhakrishnan introduced the notion of essential covers. An essential cover is a collection of hyperplanes that form a minimal cover of the vertices of the hypercube, and every coordinate is influential in at least one of the hyperplanes. Linial and Radhakrishnan proved using algebraic tools that every essential cover of the n-cube must be of size at least  \(\Omega (\sqrt{n})\) $\mathrm{\Omega }(\sqrt{n})$ . We devise a stronger lower bound method, and show that the size of every essential cover is at least  \(\Omega (n^{0.52})\) $\mathrm{\Omega }(n^{0.52})$ . This result has implications in proof complexity, because essential covers have been used to prove lower bounds for several proof systems.