  We prove that for a large family of product graphs, and for Kneser graphs
$K(n,\alpha n)$ with fixed $\alpha <1/2$, the following holds. Any set of
vertices that spans a small proportion of the edges in the graph can be made
independent by removing a small proportion of the vertices of the graph. This
allows us to strengthen the results of [DinurFR06] and [DinurF09], and show
that any independent set in these graphs is almost contained in an independent
set which depends on few coordinates. Our proof is inspired by, and follows
some of the main ideas of, Fox's proof of the graph removal lemma [Fox11].
