Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function f : N → N we construct a χ-bounded hereditary class of graphs C with the property that for every integer n ≥ 2 there is a graph in C with clique number at most n and chromatic number at least f (n). In particular, this proves that there are hereditary classes of graphs that are χ-bounded but not polynomially χ-bounded.

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