On the Nonuniform Fisher Inequality

Let F be a family of m subsets (lines) of a set of n elements (points). Suppose that each pair of lines has λ points in common for some positive λ. The Nonuniform Fisher Inequality asserts that under these circumstances m ⩽ n. We examine the case when m = n. We give a short proof of the fact that (with the exception of a trivial case) such an F must behave like a geometry in the following sense: a line must pass through each pair of points. This generalizes a result of de Bruijn and Erdos.