Extremal Hyperplane Problems on the Hypercube
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xu-zixuanxu-phd-math-2026-thesis.pdf
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Author(s)
Xu, Zixuan
Advisor(s)
Sauermann, Lisa
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
The n-dimensional hypercube in R n has vertex set {0, 1}^n and edges between two vertices x, y ∈ {0, 1}^n if they differ in exactly one coordinate. In this thesis, we study extremal questions of hyperplane collections with respect to the hypercube in two directions. The first direction considers covering every vertex of the hypercube using hyperplanes. A collection of (affine) hyperplanes H in R n is a cover of the n-dimensional hypercube {0, 1}^n if for every vertex v ∈ {0, 1}^n , there exists a hyperplane H ∈ H such that v ∈ H. We consider the problem of estimating the minimum size of a hyperplane cover. There is a trivial answer to this question, namely two hyperplanes x_1 = 1 and x_1 = 0 suffice. However, such an answer is not satisfactory since such a hyperplane cover is degenerate in the sense that none of the other variables appear in any of the hyperplane equations. Therefore, it is natural to impose certain nondegeneracy condition on the hyperplane cover. One type of nondegenerate covers is called skew covers where the normal vector of every hyperplane in the collection must have full support. We obtain an asymptotically tight lower bound for the minimum size of a skew cover up to constant factors. Furthermore, we strengthen our lower bound result to a weaker nondegeneracy condition. Another well-studied type of hyperplane cover is called essential covers, where every variable must appear with nonzero coefficient in at least one of the hyperplanes and every hyperplane must cover a vertex that is not covered by any other hyperplane. We prove that any essential cover must contain at least Ω(n^2/3 / (log n) ^2/3 ) hyperplanes, improving on previous results of Linial–Radhakrishnan, Yehuda–Yehudayoff and Araujo–Balogh–Mattos.
The second direction, closely related to the hyperplane cover problem, is to estimate the minimum size of a hyperplane collection that slices every edge of the hypercube. We say that a hyperplane H in R^n slices an edge of the n-dimensional hypercube if H contains exactly one interior point of the edge. Equivalently, for an edge vv′ with endpoints v, v′ ∈ {0, 1}^n , the hyperplane H = {x ∈ R^n | ⟨a, x⟩ = b} slices the edge vv′ if ⟨a, v⟩ − b and ⟨a, v′ ⟩ − b are both non-zero and have different signs. It is widely conjectured that Ω(n) hyperplanes are required to slice every edge of the hypercube. We confirm the conjecture in the case where the hyperplanes have bounded integer coefficients {−C, . . . , C} for some constant C > 0, generalizing a result of Alon–Bergmann–Coppersmith–Odlyzko. Furthermore, we prove a new lower bound of Ω(n^13/19 · log ^−32/19 n) in the general setting, improving on previous results on O’Neil, Yehuda–Yehudayoff and Klein.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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