Standard Monomials and Extremal Vector Systems

Tamás Mészáros, Lajos Rónyai

Research output: Contribution to journalArticle

Abstract

A set system F⊆2[n] shatters a given set S⊆[n] if 2S={F∩S:F∈F}. The Sauer-Shelah lemma states that in general, F shatters at least |F| sets. A set sytstem is called shattering-extremal if it shatters exactly |F| sets. In [Mészáros, T., “S-extremal set systems and Gröbner bases”, Diploma Thesis, Budapest University of Technology and Economics, 2010] and [Mészáros, T., L. Rónyai, “Some combinatorial application of Gröbner bases”, In: F. Winkler, Algebraic Informatics, CAI 2011, Lecture Notes in Computer Science 6742, Springer 2011, 64–83] an algebraic characterization of shattering-extremal set systems was given, which offered the possibility to generalize the notion of extremality to general finite vector systems. Here we generalize the results obtained for set systems to this more general setting, and as an application, strengthen a result of Dong, Li and Zhang from [Dong, T., Z. Li, S. Zhang, Finite sets of affine points with unique associated monomial order quotient bases, Journal of Algebra and its Applications 11(2) (2012), 1250025].

Original languageEnglish
Pages (from-to)855-861
Number of pages7
JournalElectronic Notes in Discrete Mathematics
Volume61
DOIs
Publication statusPublished - Aug 1 2017

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Set Systems
Computer aided instruction
Computer science
Algebra
Generalise
Monomial
Finite Set
Lemma
Quotient
Computer Science
Economics
Standards

Keywords

  • extremal vector systems
  • Gröbner bases
  • shattering-extremal set systems
  • standard monomials

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

Standard Monomials and Extremal Vector Systems. / Mészáros, Tamás; Rónyai, Lajos.

In: Electronic Notes in Discrete Mathematics, Vol. 61, 01.08.2017, p. 855-861.

Research output: Contribution to journalArticle

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