Kneser graphs are like Swiss cheese

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].