Consider the graph defined by
Clearly, . By hypothesis, we have for at least pairs . By Lemma 10.2, this implies that . Applying Lemma 10.3, we conclude that there exists a subset with and . Applying Corollary 13.40, we may find a subspace such that and a subset of cardinality at most such that , where . If we let be as in Lemma 9.2, this implies on taking projections the projection of to is covered by at most translates of . This implies that
since , we conclude that
By hypothesis, is covered by at most translates of , and hence by at most translates of . As is a homomorphism, each such translate can be written in the form for some . The number of translates is bounded by
By the pigeonhole principle, one of these translates must then contain at least elements of (and hence of ), and the claim follows. □