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Given positive integers e1, e2 , let Xi denote the set of ei -dimensional subspaces of a fixed finite vector space V=(Fq)e1+e2 . Let Yi be a non-empty subset of Xi and let αi= | Yi| / | Xi| . We give a positive lower bound, depending only on α1, α2, e1, e2, q , for the proportion of pairs (S1, S2) ∈ Y1× Y2 which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.
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