  A class $\mathcal F$ of graphs is $\chi$-bounded if there is a function $f$
such that $\chi(H)\le f(\omega(H))$ for all induced subgraphs $H$ of a graph in
$\mathcal F$. If $f$ can be chosen to be a polynomial, we say that $\mathcal F$
is polynomially $\chi$-bounded. Esperet proposed a conjecture that every
$\chi$-bounded class of graphs is polynomially $\chi$-bounded. This conjecture
has been disproved; it has been shown that there are classes of graphs that are
$\chi$-bounded but not polynomially $\chi$-bounded. Nevertheless, inspired by
Esperet's conjecture, we introduce Pollyanna classes of graphs. A class
$\mathcal C$ of graphs is Pollyanna if $\mathcal C\cap \mathcal F$ is
polynomially $\chi$-bounded for every $\chi$-bounded class $\mathcal F$ of
graphs. We prove that several classes of graphs are Pollyanna and also present
some proper classes of graphs that are not Pollyanna.
