  A hole in a graph $G$ is an induced cycle of length at least four, and a
$k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is
well known that if $G$ does not contain any holes then its chromatic number is
equal to its clique number. In this paper we show that, for any $k$, if $G$
does not contain a $k$-multihole, then its chromatic number is at most a
polynomial function of its clique number. We show that the same result holds if
we ask for all the holes to be odd or of length four; and if we ask for the
holes to be longer than any fixed constant or of length four. This is part of a
broader study of graph classes that are polynomially $\chi$-bounded.
