The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers e₁, e₂ , let X_i denote the set of e_i -dimensional subspaces of a fixed finite vector space V=(Fq)e1+e2 . Let Y_i be a non-empty subset of X_i and let α_i= | Y_i| / | X_i| . 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.