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

A short proof that $\chi$ can be bounded $\epsilon$ away from $\Delta+1$ towards $\omega$