Dp-rank and forbidden configurations

TL;DR

This paper establishes a connection between the model-theoretic concept of dp-rank and the combinatorial properties of VC classes, showing that a theory's dp-rank corresponds to interpreting certain maximum VC classes.

Contribution

It provides a new characterization of dp-rank via the interpretation of maximum VC classes, linking model theory and combinatorial geometry.

Findings

Dp-rank corresponds to interpreting maximum VC classes.

A theory has an ICT pattern of depth k iff it interprets a k-maximum VC class.

Bridges model theory with combinatorial properties of VC classes.

Abstract

A theory $T$ is shown to have an ICT pattern of depth $k$ in variables $\xbar$ iff it interprets some $k$-maximum VC class in $|\xbar|$ parameters.