extremal set systems
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2020 ◽  
Vol 276 ◽  
pp. 92-101
Author(s):  
Christopher Kusch ◽  
Tamás Mészáros

10.37236/4548 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=\{F~\cap~S ~:~F~\in~\mathcal{F}\}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.


2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai

We say that a set system ℱ⊆2[n] shatters a given set S⊆[n] if 2S={F∩S:F∈ℱ}. The Sauer inequality states that in general, a set system ℱ shatters at least |ℱ| sets. Here, we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |ℱ| 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.


2004 ◽  
Vol 105 (1) ◽  
pp. 143-159 ◽  
Author(s):  
Zoltán Füredi ◽  
Benny Sudakov

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