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