  The {\em independence ratio} of a graph $G$ is defined by \[ \iota(G) :=
\sup_{X \subset V(G)} \frac{|X|}{\alpha(X)},\] where $\alpha(X)$ is the
independence number of the subgraph of $G$ induced by $X$. The independence
ratio is a relaxation of the chromatic number $\chi(G)$ in the sense that
$\chi(G) \geq \iota(G)$ for every graph $G$, while for many natural classes of
graphs these quantities are almost equal. In this paper, we address two old
conjectures of Erd\H{o}s on cycles in graphs with large chromatic number and a
conjecture of Erd\H{o}s and Hajnal on graphs with infinite chromatic number.
