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有关的相当普遍的相关问题。切尔林将研究几个代表相当广泛和知之甚少的领域。具体而言，将考虑以下问题。(1)一般无三角形图是否为伪有限。(2)对于不含3阶传递竞赛的大有限标记有向图是否存在0 - 1定律？(3)对于由花环积Sym(n)*Sym(d)作用于k集的d元组所给出的结构，允许量词消除的语言的最小复杂性是什么？(4)在对三阶子结构施加负约束的假设下，具有三个对称2型语言的无限本原齐次结构的分类是什么？合作研究者托马斯将继续研究omega-分类结构的缩小和扩大。他还计划研究-稳定群作用和无限对称群的独立性结果。数学逻辑中的模型论子领域关注公理系统，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而不真正考虑其内容来回答。令人惊讶的是，有时通过这种方式可以学到很多东西，因为它往往似乎是不劳而获。模型理论家已经发展了一个丰富的主题，并且能够阐明代数中诸如群理论和场理论等研究得很好的主题。

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.