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Discrete Math

Hypergraphs and higher dimension analogs of VC dimension

Henry Towsner, Penn

Location:  Hill 705
Date & time: Monday, 19 March 2018 at 2:00PM - 3:00PM

Abstract:  The notion of bounded VC dimension is a property at the intersection of combinatorics and probability.  This family has been discovered repeatedly and studied from various perspectives - for instance, in model theory, theories with bounded VC dimension are known as NIP (the theories which do Not have the Independence Property).  One useful property is that graphs with bounded VC dimension are the graphs that can be always be finitely approximated in a "random-free" way: graphs with bounded VC dimension satisfy a strengthening of Szemeredi's Regularity Lemma in which the densities between the pieces of the partition are either close to 0 or close to 1.

 The generalization of VC dimension to higher arity, known in model theory as k-NIP for various k, has been less well-studied.  We summarize some known facts about this generalization, including a new result (joint with Chernikov) showing k-NIP hypergraphs have a similar kind of random-free approximation.  To define this notion cleanly, we will need to use a measure-theoretic approach to working with hypergraph regularity.

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