# Shatter functions with polynomial growth rates

## TL;DR

This paper investigates how a specific value of the shatter function influences the overall growth rate of set systems, challenging existing conjectures and expanding understanding of combinatorial set theory.

## Contribution

It provides new insights into the relationship between shatter function values and growth rates, and refutes a conjecture extending Sauer's Lemma.

## Key findings

- Refutes a conjecture of Bondy and Hajnal
- Establishes bounds on growth rates based on shatter function values
- Enhances understanding of polynomial growth in set systems

## Abstract

We study how a single value of the shatter function of a set system restricts its asymptotic growth. Along the way, we refute a conjecture of Bondy and Hajnal which generalizes Sauer's Lemma.