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DOP and FCP in generic structures

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science M/C 249, University of Illinois at Chicago, 851 S. Morgan, Chicago, Illinois 60607, USA, E-mail: jbaldwin@uic.edu Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
Saharon Shelah
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, E-mail: shelah@sunrise.huji.ac.il

Extract

We work throughout in a finite relational language L. This paper is built on [2] and [3]. We repeat some of the basic notions and results from these papers for the convenience of the reader but familiarity with the setup in the first few sections of [3] is needed to read this paper. Spencer and Shelah [6] constructed for each irrational α between 0 and 1 the theory Tα as the almost sure theory of random graphs with edge probability n−α. In [2] we proved that this was the same theory as the theory Tα built by constructing a generic model in [3]. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that Tα has the dimensional order property and does not have the finite cover property.

We work in the framework of [3] so probability theory is not needed in this paper. This choice allows us to consider a wider class of theories than just the Tα. The basic facts cited from [3] were due to Hrushovski [4]; a full bibliography is in [3]. For general background in stability theory see [1] or [5].

We work at three levels of generality. The first is given by an axiomatic framework in Context 1.10. Section 2 is carried out in this generality. The main family of examples for this context is described in Example 1.3. Sections 3 and 4 depend on a function δ assigning a real number to each finite L-structure as in these examples. Some of the constructions in Section 3 (labeled at the time) use heavily the restriction of the class of examples to graphs. The first author acknowledges useful discussions on this paper with Sergei Starchenko.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1998

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References

[1] Baldwin, J. T., Fundamentals of stability theory, Springer-Verlag, 1988.Google Scholar
[2] Baldwin, J. T. and Shelah, S., Randomness and semigenericity, to appear in Transactions of the American Mathematical Sorefy.Google Scholar
[3] Baldwin, J. T. and Shi, Niandong, Stable generic structures, Annals of Pure and Applied Logic, vol. 79 (1996), pp. 135.Google Scholar
[4] Hrushovski, E., A stable ℵ0-categoricalpseudoplane, preprint, 1988.Google Scholar
[5] Shelah, S., Classification theory and the number of nonisomorphic models, second ed., North-Holland, 1991.Google Scholar
[6] Shelah, S. and Spencer, J., Zero-one laws for sparse random graphs, Journal of American Mathematical Society, vol. 1 (1988), pp. 97115.Google Scholar
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