Submission date: 25 May 2020

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 coordinatisation [14] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language L and any positive integer d the class C(L,d) of all finite L-structures with at most d 4-types is a polynomial exact class in L, where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.

Mathematics Subject Classification:

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Full text arXiv 2005.12341: pdf, ps.