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

通过推广随机图的构造，得到了一般结构的概念。可以这样做。设K是一类有限结构。我们假设K上有一个量纲函数δ我们还假设对于δ，K具有合并性质。然后，通过相当随机地合并K中的结构，我们可以得到一个无限（可数）结构。这个无限结构M被称为k型结构。M由以下两个性质表征：(1)M的每一个有限子结构与K中的一个元素同构，(2)如果a < K中的B，且a <M，则M有B在a上的同构副本。第一个条件可以用句子表述，但不幸的是，第二个条件不能用句子表达。这是因为关系A<M不能用一个句子来表达。引入了强合并性质的概念，证明了如果K类具有强合并性质，则K-一般模型的理论是公化的。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.