Essential covers of the cube by hyperplanes

Nathan Linial*, Jaikumar Radhakrishnan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

A set L of linear polynomials in variables X1, X2, ..., Xn with real coefficients is said to be an essential cover of the cube {0, 1}n if (E1) for each ν ∈ {0, 1}n, there is a p ∈ L such that p (ν) = 0; (E2) no proper subset of L satisfies (E1), that is, for every p ∈ L, there is a ν ∈ {0, 1}n such that p alone takes the value 0 on v; (E3) every variable appears (in some monomial with non-zero coefficient) in some polynomial of L. Let e(n) be the size of the smallest essential cover of {0, 1}n. In the present note we show that 1/2 (√4n+1 + 1) le; e(n) ≤ ⌈n/2⌉ + 1.

Original languageAmerican English
Pages (from-to)331-338
Number of pages8
JournalJournal of Combinatorial Theory. Series A
Volume109
Issue number2
DOIs
StatePublished - Feb 2005

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