Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 126214, 8 pages
Research Article

Shattering-Extremal Set Systems of Small VC-Dimension

1Department of Mathematics, Central European University, Nádor u. 9, 1051 Budapest, Hungary
2Computer and Automation Research Institute, Hungarian Academy of Sciences, 1518 Budapest, P.O. Box 63, Hungary
3Institute of Mathematics, Budapest University of Technology and Economics, 1521 Budapest, P.O. Box 91, Hungary

Received 29 October 2012; Accepted 17 December 2012

Academic Editors: E. Bannai, C. da Fonseca, A. P. Godbole, E. Kiliç, and B. Wu

Copyright © 2013 Tamás Mészáros and Lajos Rónyai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We say that a set system shatters a given set if . The Sauer inequality states that in general, a set system shatters at least sets. Here, we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also, from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections.