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.
