Given positive integers \(e_1,e_2\), let \(X_i\) denote the set of \(e_i\)-dimensional subspaces of a fixed finite vector space\(V=({\mathbb F}_q)^{e_1+e_2}\). Let \(Y_i\) be a non-empty subset of \(X_i\) and let \(\alpha _i = |Y_i|/|X_i|\). We give a positive lower bound, depending only on \(\alpha _1,\alpha _2,e_1,e_2,q\), for the proportion of pairs \((S_1,S_2)\in Y_1\times Y_2\) 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.

给定正整数\(e_1,e_2\)，让\(X_i\)表示固定有限向量空间\(V=({\mathbb F}_q)^{e_1 的 \( e_i\)维子空间的集合+e_2}\)。令\(Y_i\)为\(X_i\)的非空子集并令\(\alpha _i = |Y_i|/|X_i|\)。我们给出一个正下界，仅取决于\(\alpha _1,\alpha _2,e_1,e_2,q\)，对于对的比例\((S_1,S_2)\in Y_1\times Y_2\)相交平凡。作为一个应用，我们限制了有限经典空间中平凡相交的非退化子空间对的比例。这个问题的动机是识别经典群的算法。通过使用代数图论中的技术，我们能够处理 2 阶域上的正交群，这是 Niemeyer、Praeger 和第一作者未能解决的情况。