Dirichlet Sets and Erdös-kunen-mauldin Theorem

Dirichlet Sets, Erdős-kunen-mauldin Theorem, and Analytic Subgroups of the Reals

We prove strengthenings of two well-known theorems related to the Lebesgue measure and additive structure of the real line. The first one is a theorem of Erdős, Kunen, and Mauldin stating that for every perfect set there exists a perfect set of measure zero such that their algebraic sum is the whole real line. The other is Laczkovich’s theorem saying that every proper analytic subgroup of the r...

Extremal Matroid Theory and The Erdös-Pósa Theorem

The Di erences Between Kurepa Trees And Jech

By an !1{tree we mean a tree of power !1 and height !1. An !1{tree is called a Kurepa tree if all its levels are countable and it has more than !1 branches. An !1{tree is called a Jech{Kunen tree if it has branches for some strictly between !1 and 2 !1 . In x1, we construct a model of CH plus 21 > !2, in which there exists a Kurepa tree with no Jech{Kunen subtrees and there exists a Jech{Kunen ...

On Possible Non-homeomorphic Substructures of the Real Line

We consider the problem, raised by Kunen and Tall, of whether the real continuum can have non-homeomorphic versions in different submodels of the universe of all sets. This requires large cardinals, and we obtain an exact consistency strength: Theorem 1. The following are equiconsistent : (i) ZFC + ∃κ a Jónsson cardinal ; (ii) ZFC + ∃M a sufficiently elementary submodel of the universe of sets ...

Operator Algebras and Mauldin Williams Graphs

We describe a method for associating a C-correspondence to a Mauldin-Williams graph and show that the Cuntz-Pimsner algebra of this Ccorrespondence is isomorphic to the C-algebra of the underlying graph. In addition, we analyze certain ideals of these C-algebras. We also investigate Mauldin-Williams graphs and fractal C-algebras in the context of the Rieffel metric. This generalizes the work of...