Essential covers of the hypercube require many hyperplanes (2310.05775v1)

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.