Problems in Combinatorial Set Theory

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I propose a program of research in the general area of combinatorial set theory, with an emphasis on "singular cardinalcombinatorics" in the style of Shelah. An example of the sort of questions that I propose to work on is the question (due to Woodin)whether failure of SCH at a singular cardinal implies the existence ofan Aronszajn tree at the successor of that cardinal. I am interested inobtaining ZFC results as well as complementary consistency results: the methods that I plan to use include PCF theory, inner models, Radin forcingand some recent generalisations of proper forcing. Combinatorial set theory can be seen as an attempt to take some veryelementary mathematical ideas (counting, ordering, permuting) andmake sense of them in the context of infinite sets. Typical problems mightinvolve finding an infinite path through an infinite tree, or colouring the vertices in an infinite graph so that no two adjacent vertices have the same colour. Problems of this kind turn out to be surprisinglydeep and have found applications in many areas of mathematics. The focusof my proposal is problems involving so-called "singular cardinals",which (by some measure) are the hardest infinite sets to understand.