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Algebra universalis

, 79:2 | Cite as

A note on linear Sperner families

  • Gábor Hegedűs
  • Lajos Rónyai
Article
Part of the following topical collections:
  1. In memory of E. Tamás Schmidt

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 \(\mathbf {v}\in \{0,1\}^n\) 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 \(a_1v_1+\cdots +a_nv_n=k\), where the \(a_i\) and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that \(0<a_1\le a_2\le \cdots \le a_n\). As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.

Keywords

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

Mathematics Subject Classification

Primary 13P25 Secondary 13P10 05D05 

Notes

Acknowledgements

We thank the referee for valuable suggestions.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Antal Bejczy Center for Intelligent RoboticsÓbuda UniversityBudapestHungary
  2. 2.Institute of Computer Science and ControlHungarian Academy of SciencesBudapestHungary
  3. 3.Department of AlgebraBudapest University of Technology and EconomicsBudapestHungary

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