# Thicket Density

## TL;DR

This paper introduces a new shatter function for set systems, linking its growth rate to Shelah's 2-rank, offering a novel perspective beyond VC dimension in combinatorial set theory.

## Contribution

It defines a new type of shatter function with a Sauer-Shelah dichotomy, connecting polynomial growth to Shelah's 2-rank instead of VC dimension.

## Key findings

- Established a new shatter function satisfying a Sauer-Shelah dichotomy.
- Linked polynomial growth of the shatter function to Shelah's 2-rank.
- Provided a new framework for understanding growth rates in set systems.

## Abstract

We define a new type of "shatter function" for set systems that satisfies a Sauer-Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah's 2-rank instead of VC dimension. We identify the rate of growth of this shatter function, the quantity analogous to VC density, with Shelah's $\omega$-rank.