随机图上的极小函数
Minimal functions on the random graph
摘要：
我们证明了随机图上存在一个由14个非平凡有限函数组成的系统，该系统具有以下性质：随机图上的任何非平凡函数通过与自同构的组合和拓扑闭包生成该系统的一个函数，并且该系统在系统的任何子集都不具有相同性质的意义下是最小的。该定理是通过证明随机图的有限幂元组的着色的拉姆齐型定理，并通过应用此定理来找到随机图上任何函数的行为中的规则模式而获得的。作为模型论的推论，我们重新推导了Simon Thomas关于随机图的一阶闭约简的分类定理，并证明了该定理的一些改进;同时，我们得到了在本原正定义下闭的最小约简的分类，并证明了随机图的所有约简都是模型完备的.
英文摘要：
We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by applying this to find regular patterns in the behavior of any function on the random graph. As model-theoretic corollaries of our methods we re-derive a theorem of Simon Thomas classifying the first-order closed reducts of the random graph, and prove some refinements of this theorem; also, we obtain a classification of the minimal reducts closed under primitive positive definitions, and prove that all reducts of the random graph are model-complete.
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