On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

TL;DR

This paper extends hyperplane covering bounds to symmetric and blockwise symmetric subsets of the hypercube, using polynomial methods, and explores the limits of this approach for symmetry-preserving covers.

Contribution

It introduces higher multiplicity hyperplane covers for symmetric hypercube subsets and demonstrates that polynomial methods suffice for these cases, highlighting their limitations.

Findings

Solved hyperplane covering problem for symmetric sets with higher multiplicities.

Showed polynomial method is sufficient for symmetric and blockwise symmetric sets.

Identified the approach's limitations for more general symmetry-preserving covers.

Abstract

Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem.