  We prove that every graph $G$ for which $\omega(G) \geq 3/4(\Delta(G) + 1)$,
has an independent set $I$ such that $\omega(G - I) < \omega(G)$. It follows
that a minimum counterexample $G$ to Reed's conjecture satisfies $\omega(G) <
3/4(\Delta(G) + 1)$ and hence also $\chi(G) > \lceil 7/6\omega(G) \rceil$. We
also prove that if for every induced subgraph $H$ of $G$ we have $\chi(H) \leq
\max{\lceil 7/6\omega(H) \rceil, \lceil \frac{\omega(H) + \Delta(H) +
1}{2}\rceil}$, then we also have $\chi(G) \leq \lceil \frac{\omega(G) +
\Delta(G) + 1}{2}\rceil$. This gives a generic proof of the upper bound for
line graphs of multigraphs proved by King et al.
