Geodesic
Characterizations of Finite Projective and Affine Spaces
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 64-75

Voir la notice de l'article provenant de la source Cambridge University Press

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.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 to4 if and only if there are positive integers v, k, and y, with μ > 1 and (μ – l)(v — k) ≠ (k — μ)2such that the following assumptions hold. (I) Every block is on k points, and every two intersecting blocks are on μ 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 > μ. 
Characterizations of Finite Projective and Affine Spaces. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 64-75. doi: 10.4153/CJM-1969-007-9
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