On the abstract chromatic number and its computability for finitely axiomatizable theories

TL;DR

This paper extends the concept of an abstract chromatic number to graphs with extra structure, providing a concrete, computable formula for it in finitely axiomatizable theories, and characterizes the maximum density of cliques.

Contribution

It introduces a concrete, computable formula for the abstract chromatic number in finitely axiomatizable theories, extending previous asymptotic density results.

Findings

Extended the asymptotic density characterization to t-cliques.

Provided a concrete, computable formula for the abstract chromatic number.

Utilized a partite Ramsey's Theorem for structures.

Abstract

The celebrated Erd\H{o}s--Stone--Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1 - 1/(\chi(\mathcal{F})-1) + o(1)$, where $\chi(\mathcal{F})$ is the minimum chromatic number of a graph in $\mathcal{F}$. In Examples 25 and 31 of [L. N. Coregliano and A. A. Razborov. Semantic limits of dense combinatorial objects. Uspekhi Mat. Nauk, 75(4(454)):45-152, 2020], it was shown that this result can be extended to the general setting of graphs with extra structure: the maximum asymptotic density of a graph with extra structure without some induced subgraphs is $1 - 1/(\chi(I) - 1) + o(1)$ for an appropriately defined abstract chromatic number $\chi(I)$. As the name suggests, the original formula for the abstract chromatic number is so abstract that its (algorithmic) computability was left open. In this paper, we both extend this result to characterize maximum asymptotic density of $t$-cliques in of graphs with extra structure without some induced subgraphs in terms of $\chi(I)$ and we present a more concrete formula for $\chi(I)$ that allows us to show its computability when both the extra structure and the forbidden subgraphs can be described by a finitely axiomatizable universal first-order theory. Our alternative formula for $\chi(I)$ makes use of a partite version of Ramsey's Theorem for structures on first-order relational languages.