A note on a very abstract chromatic number and extremal problems

TL;DR

This paper explores the abstract chromatic number's role in extremal graph theory, generalizing its application to other graph parameters and demonstrating these ideas with two examples.

Contribution

It extends the concept of the abstract chromatic number to other extremal functions, providing generalizations and concrete examples.

Findings

The abstract chromatic number influences Turán number asymptotics.

Generalizations apply to other extremal graph parameters.

Two specific examples demonstrate these generalizations.

Abstract

The abstract chromatic number was introduced by Razborov and Coregliano in 2020 in using the language of model theory, and was used to extend the Erd\H os-Stone-Simonovits theorem to graphs with extra structures. A purely combinatorial version was introduced by Gerbner, Hama Karim and Kucheriya in 2026, who also showed that in addition to the asymptotic bound on the Tur\'an number, the abstract chromatic number determines the asymptotics of several other Tur\'an-type functions. We observe that the chromatic number is used here due to its special role in determining the asymptotics of the Tur\'an number. For other extremal functions, other graph parameters may play a similar role and let us extend results in a similar fashion. We prove the appropriate generalizations and show two examples where this happens.