  In 1998 the second author proved that there is an $\epsilon>0$ such that
every graph satisfies $\chi \leq \lceil
(1-\epsilon)(\Delta+1)+\epsilon\omega\rceil$. The first author recently proved
that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set
intersecting every maximum clique. In this note we exploit the latter result to
give a much shorter, simpler proof of the former. We include, as a certificate
of simplicity, an appendix that proves all intermediate results with the
exception of Hall's Theorem, Brooks' Theorem, the Lov\'asz Local Lemma, and
Talagrand's Inequality.
