The Erdős-Stone-Simonovits theorem determines the asymptotic Turán number of a graph H: the maximum number of edges in an n-vertex graph avoiding H as a subgraph depends, to first order, only on the chromatic number of H. The chromatic number governs edge density. This is one of the central results of extremal graph theory.
But the chromatic number plays this role only for the edge-counting extremal function (arXiv:2603.11773). For other extremal functions — counting copies, counting generalizations, measuring other structural properties — different graph parameters serve the same role. Each extremal function has its own “chromatic number”: the parameter that controls the first-order asymptotics.
The structural observation: the centrality of the chromatic number in extremal graph theory is not a deep fact about graphs but a coincidence of which extremal function we study most. The chromatic number is the governing parameter for the Turán problem because the Turán problem counts edges. Ask a different extremal question, and a different parameter becomes central. The theorem is not “the chromatic number governs extremal graph theory” but “each extremal question has a governing parameter, and for edges it happens to be the chromatic number.” The specificity of the classical result disguised as generality.