Dilip Raghavan

Cofinal types of ultrafilters

Abstract We study Tukey types of ultrafilters on ω , focusing on the question of when Tukey reducibility is equivalent to Rudin–Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ ω 1 ] ω . We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.

There is a Van Douwen MAD family

We answer a long-standing question of Van Douwen by proving in ZFC that there is a MAD family of functions in ω ω that is also maximal with respect to infinite partial functions. In Section 3 we apply the idea of trace introduced in this proof to the still open question of whether analytic MAD families exist in ω ω . Using the idea of trace, we show that any analytic MAD families that may exist in ω ω must satisfy strong combinatorial constraints. We also show that it is consistent to have MAD families in ω ω that satisfy these constraints.

Bounding, splitting, and almost disjointness

Abstract We investigate some aspects of bounding, splitting, and almost disjointness. In particular, we investigate the relationship between the bounding number, the closed almost disjointness number, the splitting number, and the existence of certain kinds of splitting families.

Almost Disjoint Families and Diagonalizations of Length Continuum

We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

Suslin trees, the bounding number, and partition relations

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)2 for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ+ ↛ (κ+, logκ(κ+) + 2)2. The consistency of the positive partition relation b → (b, α)2 for all α < ω1 for the bounding number b is also established from large cardinals.

Weakly Represented Families in Reverse Mathematics

We study the proof strength of various second order logic principles that make statements about families of sets and functions. Usually, families of sets or functions are represented in a uniform way by a single object. In order to be able to go beyond the limitations imposed by this approach, we introduce the concept of weakly represented families of sets and functions. This allows us to study various types of families in the context of reverse mathematics that have been studied in set theory before. The results obtained witness that the concept of weakly represented families is a useful and robust tool in reverse mathematics.