Let Γ be the unique (up to isomorphism) countable graph with the following property: (*) Given any two finite disjoint subsets U and V of Γ, there exists a vertex z ∈ Γ joined to every vertex in U and to none in V.Thus Γ is the countable, universal, homogeneous graph; also known as the random graph. In this paper, we shall study the reducts of Γ Here a reduct of Γ is defined to be a permutation group (G, Γ) such that:(i) Aut(Γ) ≤ G; and(ii) G is a closed subgroup of Sym(Γ).Equivalently, there exists a structure for some language L such that:(iii) has universe Γ;(iv) for each R ∈ L, is definable without parameters in Γ; and(v) G = Aut().

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1. The theme P:={g€S\l ig = I , for all i} = ITS,.If G t is the projection of G 0 P into S t then G n P < IlGi. Noŵ

Let IF be a finite field, and let (lP, B) be a nontrivial 2-(n, k, 1)-design over F. Then each point a E ~ induces a (k -1)-spread S~ on ~/c~. (]P, B) is said to be a geometric design ifS,~ is a geometric spread on ~/c~ for each c~ E P. In this paper, we prove that there are no geometric designs over any finite field IF.Mathematics Subject Classifications (1991): 51E20, 05B05, 51E14.

We prove that for each n ≥ 1 n \geq 1 , the classification problem for torsion-free abelian groups of rank n + 1 n+1 is not Borel reducible to that for torsion-free abelian groups of rank n n .