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

The celebrated Erdős–Stone–Simonovits theorem characterizes the asymptotic maximum edge density in F-free graphs as 1−1/(χ(F)−1)+o(1), where χ(F) is the minimum chromatic number of a graph in F. In [6, Examples 25 and 31], it was shown that this result can be extended to the general setting of graphs with extra structure: the asymptotic maximum edge density of a graph with extra structure without some induced subgraphs is 1−1/(χ(I)−1)+o(1) for an appropriately defined abstract chromatic number χ(I). As the name suggests, the original formula for the abstract chromatic number is so abstract that its (algorithmic) computability was left open.