A note on linear Sperner families

Gábor Hegedűs, Lajos Rónyai

Research output: Contribution to journalArticle

Abstract

In an earlier work we described Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors v∈ { 0 , 1 } n of the complete d uniform set family over the ground set [n]. In particular, it turns out that the standard monomials of the above ideal are ballot monomials. We give here a partial extension of this fact. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation a1v1+ ⋯ + anvn= k, where the ai and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that 0 < a1≤ a2≤ ⋯ ≤ an. As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.

Original languageEnglish
Article number2
JournalAlgebra Universalis
Volume79
Issue number1
DOIs
Publication statusPublished - Mar 1 2018

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Keywords

  • Ballot monomial
  • Characteristic vector
  • Gröbner basis
  • Polynomial function
  • Shattering
  • Sperner family
  • Standard monomial

ASJC Scopus subject areas

  • Algebra and Number Theory

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