Characterizations of Finite Projective and Affine Spaces

THEOREM 1. A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to 4 if and only if there are positive integers v, k, and y, with ju > 1 and (/A — l)(v — k) 7* (k — ju) such that the following assumptions hold. (I) Every block is on k points, and every two intersecting blocks are on p. common points. (II) Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks. (III) Given two distinct points p and q, there is a block on p but not on q. (IV) There are v points, and v — 2 ^ k > /JL.