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Characterizations of Finite Projective and Affine Spaces

Published online by Cambridge University Press:  20 November 2018

William M. Kantor
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 64 - 75
Copyright
Copyright © Canadian Mathematical Society 1969

References

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