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# MULTIDIMENSIONAL EXACT CLASSES, SMOOTH APPROXIMATION AND BOUNDED 4-TYPES

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## Abstract

In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class ( R -mec), a special kind of multidimensional asymptotic class ( R -mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language $\mathcal {L}$ and any positive integer d the class $\mathcal {C}(\mathcal {L},d)$ of all finite $\mathcal {L}$ -structures with at most d 4-types is a polynomial exact class in $\mathcal {L}$ , where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.

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... As L is fixed, set-homogeneity ensures that there is d ∈ N such that for each M ∈ C, Aut(M ) has at most d orbits on M 4 . The result now follows from Theorem 4. 4.1 of [39], which is based on results from [14]. It is almost immediate from the definition of smooth approximation that if a family F i smoothly approximates an infinite structure M i then M i will itself be set-homogeneous. ...
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• R Kaye
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