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.

中文翻译：

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多维精确类、平滑近似和有界 4 类

结合 Anscombe、Macpherson、Steinhorn 和当前作者在 [1] 中的工作，我们研究了多维精确类的概念（R-mec)，一种特殊的多维渐近类 (R-mac) 具有产生可定义集合的精确大小的测量函数，而不仅仅是近似值。我们使用关于平滑近似 [24] 和李协调 [13] 的结果来证明以下结果（定理 4.6.4），正如 Macpherson 所猜想的：对于任何可数语言$\数学{L}$和任何正整数d班上$\mathcal {C}(\mathcal {L},d)$所有有限的$\数学{L}$- 最多有的结构d4-types 是一个多项式精确类$\数学{L}$，其中多项式精确类是具有多项式测量函数的多维精确类。