Infinite Method in Finite Model Theory and its application for aiming to solve Lachlan's conjecture.

类属结构的概念是通过推广随机图的构造而得到的。这可以如下所示：设K是一类有限结构。我们假设K上有一个维函数δ，我们还假设K关于δ具有合并性。然后，通过相当随机地合并K中的结构，我们可以得到无限(可数)的结构。这种无限结构M称为K-类属结构。M的两个性质是：(1)M的每个有限子结构与K中的一个成员同构；(2)如果A&lt；B在K中，且A&lt；M在A上有B的同构副本。第一个条件可以用句子来表示，但遗憾的是第二个条件不能用句子来表示。这是因为，A和M的关系不能用一句话来表达。引入了强合并性的概念，证明了如果K类具有强合并性，则K-通有模型的理论是公理的。这部作品是与Hirotaka Kikyo和Koichiro Ikeda合作的作品。

The notion of generic structure is obtained by generalizing the construction of random graphs. This can be done as follows.Let K be a class of finite structure. We assume there is a dimension function δ on K. We also assume that with respect to δ,K has the amalgamation property. Then by amalgamating structures in K rather randomly, we can get an inifinite (countable) strurture. This infinite structure M is called a K-generic structure. M is characterize by the following two properties (1)every finite substructure of M is isomorphic to a member in K, (2)If A<B in K, and A<M then M has an isomorphic copy of B over A.The first condition can be stated by sentences but unfortunately the second one cannot be expressed by sentences. This is because, the relation A<M cannot be expressed by a single sentence. We introduced the notion of strong amalgamation property, and prove that if the class K has the strong amalgamation property then the theory of K-generic models is axiomatized. This work is a joint work with Hirotaka Kikyo and Koichiro Ikeda.