  We outline some ongoing work related to a conjecture of Reed \cite{reed97} on
$\omega$, $\Delta$, and $\chi$. We conjecture that the complement of a
counterexample $G$ to Reed's conjecture has connectivity on the order of
$\log(|G|)$. We prove that this holds for a family (parameterized by $\epsilon
> 0$) of relaxed bounds; the $\epsilon = 0$ limit of which is Reed's upper
bound.
