Separating Polynomial $$\chi $$ -Boundedness from $$\chi $$ -Boundedness

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function \(f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}\) with \(f(1)=1\) and \(f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right) \), we construct a hereditary class of graphs \({\mathcal {G}}\) such that the maximum chromatic number of a graph in \({\mathcal {G}}\) with clique number n is equal to f(n) for every \(n\in \mathbb {N}\). In particular, we prove that there exist hereditary classes of graphs that are \(\chi \)-bounded but not polynomially \(\chi \)-bounded.