A note on a very abstract chromatic number and extremal problems

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 Turán number, the abstract chromatic number determines the asymptotics of several other Turán-type functions. We observe that the chromatic number is used here due to its special role in determining the asymptotics of the Turán 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.