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].
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