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.

扩展了 Carbonero、Hompe、Moore 和 Spirkl 最近论文中的想法，对于每个函数 \ (f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}\)和\(f (1)=1\)和\(f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right) \) ，我们构造一个遗传式图类\({\mathcal {G}}\)使得\({\mathcal {G}}\)中团数为n的图的最大色数等于f ( n ) 对于每个\( n\in \mathbb {N}\)。特别是，我们证明存在有\(\chi \)有界但不是多项式的图的遗传类\(\chi \)有界。