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The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ1-categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ1-categorical theory has either just one or just ℵ0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3.As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.

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Abstract. Let L contain only the equality symbol and let L + be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L + structures with 'edge probability' n −α . By T α , the almost sure theory of random L + -structures we mean the collection of L + -sentences which have limit probability 1. Tα denotes the theory of the generic structures for Kα (the collection of finite graphs G with δα(G) = |G| − α · | edges of G | hereditarily nonnegative). This paper unites two apparently disparate lines of research. In [8], Shelah and Spencer proved a 0-1-law for first order sentences about random graphs with edge probability n −α where α is an irrational number between 0 and 1. Answering a question raised by Lynch [5], we extend this result from graphs to hypergraphs (i.e. to arbitrary finite symmetric relational languages). Let T α denote the set of sentences with limit probability 1. The Spencer-Shelah proof proceeded by a process of quantifier elimination which implicitly showed the theories T α were nearly model complete (see below) and complete.Hrushovski in [3] refuted a conjecture of Lachlan by constructing an ℵ 0 -categorical strictly stable pseudoplane. Baldwin and Shi [1] considered a variant on his methods to construct strictly stable (but not ℵ 0 -categorical) theories T α indexed by irrational α. In this paper we show that for each irrational α, T α = T α and thus deduce that T α is not finitely axiomatizable and that T α is stable. Each T α is the theory of a 'generic' model M α of an amalgamation class K α of finite structures. Although the Hrushovski examples are easily seen to be nearly model complete this is less clear for the T α since they are not ℵ 0 -categorical. We show that each T α is nearly model complete.In the first, purely model theoretic, section of the paper we describe our basic framework and prove a sufficient condition for certain theories, including the T α , to be nearly model complete. These conditions depend upon a generalization of the notion of genericity of a structure: semigenericity, which is introduced in this paper. In the second section we consider the addition of random relations and

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