Combinatorial set theory

卡明斯提出了一个研究组合集合论的方案。要研究的领域包括多色Ramsey理论(在有限和无限环境下)，组合原理，如方块和正方形，以及具有禁止子序的不可数线性序的结构。一个统一的主题是紧凑和松散之间的紧张关系。组合集合论是对结构化集合的研究，例如图、树和排序。这些集合可以是有限的，也可以是无限的，从历史上看，有限和无限的组合数学是相当独立的；然而，最近有了惊人的应用无限方法来解决关于有限集合的问题。这一研究计划的进展应适用于集合论、逻辑、组合学和拓扑学。

Cummings proposes a program of research in combinatorial set theory. The areas to be investigated include polychromatic Ramsey theory (in both the finite and infinite settings), combinatorial principles such as diamonds and squares, and the structure of uncountable linear orderings with forbidden suborderings. A unifying theme is the tension between compactness and incompactness. Combinatorial set theory is the study of structured sets, for example graphs, trees and orderings. These sets can be finite or infinite, and historically finite and infinite combinatorics have been rather separate; however recently there have been striking applications of infinitary methods to solve problems about finite sets. Progress in this research program should find applications to set theory, logic, combinatorics and topology.