Model theory and combinatorics of banned sequences

TL;DR

This paper develops a general framework for Sauer-Shelah type lemmas, applying it to improve existing results and introduce new combinatorial bounds related to banned sequences and op-rank.

Contribution

It introduces a unified approach to prove Sauer-Shelah lemmas, improving known bounds and extending the theory to op-rank contexts.

Findings

Provided a general method for Sauer-Shelah lemmas

Improved bounds on combinatorial parameters

Extended results to op-rank setting

Abstract

We set up a general context in which one can prove Sauer-Shelah type lemmas. We apply our general results to answer a question of Bhaskar and give a slight improvement to a result of Malliaris and Terry. We also prove a new Sauer-Shelah type lemma in the context of op-rank, a notion of Guingona and Hill.