On unavoidable induced subgraphs in large prime graphs

TL;DR

This paper reestablishes that large prime graphs necessarily contain certain induced prime subgraphs, using model-theoretic methods to improve bounds and provide new structural insights.

Contribution

It offers a new proof of existing theorems on prime graphs by incorporating stability analysis and model theory, enhancing bounds and understanding.

Findings

Improved bounds on induced prime subgraphs in large prime graphs

A new structural perspective using stability and model theory

An infinitary proof supporting the finite case

Abstract

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of the same ideas, but with the key model-theoretic ingredient of first determining the so-called amount of stability of the graph. This approach changes the applicable Ramsey theorem, improves the bounds and offers a different structural perspective on the graphs in question. Complementing this, we give an infinitary proof which implies the finite result.