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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Randomness and semigenericity

Authors: John T. Baldwin and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 349 (1997), 1359-1376
MSC (1991): Primary 03C10, 05C80
MathSciNet review: 1407480
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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' $n^{-\alpha }$. By $T^{\alpha }$, the almost sure theory of random $L^+$-structures we mean the collection of $L^+$-sentences which have limit probability 1. $T_{\alpha }$ denotes the theory of the generic structures for ${\mathbf {K}} _{\alpha }$ (the collection of finite graphs $G$ with $\delta _{\alpha }(G) =|G| - \alpha \cdot |\text { edges of $G$ }|$ hereditarily nonnegative).

Theorem.. $T^{\alpha }$, the almost sure theory of random $L^+$-structures, is the same as the theory $T_{\alpha }$ of the ${\mathbf {K}} _{\alpha }$-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

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Keywords: Random graphs, 0-1-laws, stability
Received by editor(s): September 7, 1994
Additional Notes: Partially supported by NSF grant 9308768 and a visit to Simon Fraser University.
This is paper 528. Both authors thank Rutgers University and the Binational Science Foundation for partial support of this research.
Article copyright: © Copyright 1997 American Mathematical Society