可数交换群作用的描述组合学  

Descriptive combinatorics of countable abelian group actions

作　　者：高速 Su Gao

机构地区：[1]南开大学数学科学学院,天津300071

出　　处：《中国科学：数学》2024年第4期575-592,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12250710128)资助项目。

摘　　要：本文介绍描述集合论中的超有穷性问题与描述组合理论之间的关系,综述用描述集合论的方法研究可数交换群作用产生的Schreier图的组合性质所得到的结果.这些性质既包括Borel或连续染色数、边染色数和完全匹配等图论性质,也包括一般Borel作用下Borel标记集的动力系统性质.本文也介绍证明结论所采用的方法,这些方法涉及拓扑学、遍历论、几何群论和力迫法等不同数学分支.In this paper,we describe the connection between the hyperfiniteness problem in descriptive set theory and descriptive combinatorics and survey the results obtained using descriptive set theoretic methods on combinatorial properties of the Schreier graphs generated by countable abelian group actions.The properties considered in the survey include not only graph theoretic characteristics such as Borel or continuous chromatic numbers,edge chromatic numbers,and perfect matchings but also dynamic properties of Borel marker sets under Borel actions.We also summarize the methods used in our proofs,which are relevant to different branches of mathematics including topology,ergodic theory,geometric group theory,and forcing.

关 键 词：描述集合论 Borel归约 超有穷 染色数 完全匹配 标记结构 超非周期元 轨道力迫 

分 类 号：O157[理学—数学]