#####
Séminaire Lotharingien de Combinatoire, B11h (1984).

[Formerly: Publ. I.R.M.A. Strasbourg, 1985, 266/S-11, p.
120-121.]

# Bernd Voigt

# On the Evolution of Finite Affine and Projective Spaces

**Abstract.**
Let *q* be a prime power, *k*<=*m*<=*n* integers.
Choose each of the
*k*-dimensional subspaces of
(GF(*q*))^{n}
with
probability <
*\alpha*(*n*). Denote by *E* the event
that the above random set of *k*-dimensional subspaces
contains all *k*-dimensional subspaces of some
*m*-dimensional subspace. The threshold function *f*(*n*) of
*E* is
determined: if *\alpha*(*n*)/*f*(*n*)
tends
to 0 [resp. *\infty*,
nonzero constant]
then *P*(*E*) tends to 0 [resp.
1, nonzero constant]. The analogous
results for projective spaces are also obtained. The theorems
are formulated
actually for some lattices. The above results,
as well as the lattice of subsets, are all special cases.

Geschäftsführer, Lufthansa Systems Berlin GmbH

The paper has been finally published under the same title in
*IX symposium on operations research. Part I. Sections 1-4
(Osnabrück, 1984)*, pp. 313-327,
Methods Oper. Res., **49**,
Athenäum/Hain/Hanstein, Königstein, 1985.