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
Key words
descriptive set theory/Borel reducible/hyperfinite/chromatic number/perfect matching/marker structure/hyperaperiodic/orbit forcing
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