Combinatorial set theory

Cummings提出了一个研究组合集合的方案 理论要调查的领域包括多色 拉姆齐理论（在有限和无限的设置）， 组合原则，如菱形和正方形，以及 不可数禁止线性序的结构 次命令一个统一的主题是 紧性和非紧性。 组合集合论是研究结构化集合，例如 图、树和排序。这些集合可以是有限的或无限的， 在历史上，有限和无限组合学 分离;然而最近有惊人的应用程序 无穷大方法解决有限集合问题。 该研究计划的进展应该会找到可供设置的应用程序 理论、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.