Mathematical Sciences: Combinatorial Aspects of the Model Theory of Finite, Pseudofinite and Homogeneous Structures

模型中有一些相当普遍的相关问题 理论与具体的组合问题。 切尔林将研究几个代表相当广泛和差 了解地区。 特别是，以下问题将 考虑了 (1)一般的无三角形图是伪有限的。 (2)大型有限标号有向映射是否存在零一律 图不包含3阶传递竞赛图。 (3)是什么 语言的最小复杂性，允许量词 消除，对于由花环的作用给出的结构， 乘积Sym（n）*Sym（d）在k-集的d-元组上的作用？ （四） 无限本原齐次的分类是什么 一种具有三个对称2-类型的语言的结构， 负约束作用于子结构的假设 三阶。 共同研究者托马斯将继续与 研究omega范畴结构的约化和扩展。 他 还计划研究欧米茄稳定的群体行动和独立性 关于无限对称群的结果 数学逻辑中的模型论的子领域涉及 本身与公理系统，特别是关于 建立在公理系统上的理论，这些理论可以被纯粹地回答 不考虑公理的形式而不考虑真实的 他们的内容。 令人惊讶的是，有时可以学到多少东西 这样，因为它往往似乎是得到的东西， 没什么 模型理论家发展了一个丰富的学科， 在代数学中， 群论和场论一样。

There are a number of quite general related questions in model theory which are connected with concrete combinatorial problems. Cherlin will study several which represent fairly broad and poorly understood areas. In particular, the following questions will be considered. (1) Is the generic triangle-free graph pseudofinite. (2) Is there a zero-one law for large finite labelled directed graphs containing no transitive tournament of order 3. (3) What is the minimal complexity of a language, admitting quantifier elimination, for the structures given by the action of the wreath product Sym(n)*Sym(d) in its action on d-tuples of k-sets? (4) What is the classification of infinite primitive homogeneous structures for a language with three symmetric 2-types, under the hypothesis that a negative constraint is imposed on a substructure of order three. The co-investigator, Thomas, will continue related work on reducts and expansions of omega-categorical structures. He also plans to study omega-stable group actions and independence results on infinite symmetric groups. The subarea of model theory within mathematical logic concerns itself with axiomatic systems, particularly questions about theories founded on systems of axioms, which can be answered purely from considering the forms of the axioms without real regard for their content. It is surprising how much can sometimes be learned in this way, for it often seems to be getting something for nothing. Model theorists have developed a rich subject and have been able to shed light even on such well-studied topics in algebra as the theory of groups and the theory of fields.