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

在模型理论中，有许多与具体组合问题相关的非常一般的问题。切尔林将研究几个代表相当广泛和鲜为人知的领域。特别是，将考虑以下问题。(1)是一般无三角形图的伪有限。(2)不含3阶传递竞赛图的大型有限标号有向图是否存在零一定律。(3)允许量词消去的语言的最小复杂度是多少，对于由圈积Sym(N)*Sym(D)作用于k-集合的d-元组所给出的结构？(4)在三个对称2-类型的语言的假设下，在对三阶子结构施加负约束的假设下，无限本原齐次结构的分类是什么？合作研究员托马斯将继续有关欧米伽范畴结构的缩减和扩展的相关工作。他还计划研究无限对称群上的omega-稳定群作用和独立性结果。数理逻辑中的模型理论的子领域涉及公理系统，特别是关于建立在公理系统上的理论的问题，这些问题可以纯粹通过考虑公理的形式来回答，而不真正考虑它们的内容。有时可以通过这种方式学到很多东西，这是令人惊讶的，因为它往往似乎是不劳而获。模型理论家们已经发展出一门丰富的学科，甚至能够阐明诸如群论和场论等代数中研究得很好的主题。

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.