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

In 1998 the second author proved that there is an ǫ > 0 such that every graph satisfies χ ≤ ⌈(1 − ǫ)(∆ + 1) + ǫω⌉. The first author recently proved that any graph satisfying ω > 2 3 (∆+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 Lovász Local Lemma, and Talagrand’s Inequality.