  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.
