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.