### Abstract

A set system F⊆2^{[n]} shatters a given set S⊆[n] if 2^{S}={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 language | English |
---|---|

Pages (from-to) | 855-861 |

Number of pages | 7 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 61 |

DOIs | |

Publication status | Published - Aug 1 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*61*, 855-861. https://doi.org/10.1016/j.endm.2017.07.046

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

Research output: Contribution to journal › Article

*Electronic Notes in Discrete Mathematics*, vol. 61, pp. 855-861. https://doi.org/10.1016/j.endm.2017.07.046

}

TY - JOUR

T1 - Standard Monomials and Extremal Vector Systems

AU - Mészáros, Tamás

AU - Rónyai, Lajos

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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].

AB - 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].

KW - extremal vector systems

KW - Gröbner bases

KW - shattering-extremal set systems

KW - standard monomials

UR - http://www.scopus.com/inward/record.url?scp=85026733364&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85026733364&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2017.07.046

DO - 10.1016/j.endm.2017.07.046

M3 - Article

AN - SCOPUS:85026733364

VL - 61

SP - 855

EP - 861

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -