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Let $G$ be an abelian group and $A,B$ two subsets of equal size $k$ such that $A+B$ and $A+A$ both have size $2k-1$. Answering a question of Bihani and Jin, we prove that if $A+B$ is aperiodic or if there exist elements $a\in A$ and $b\in B$ such that $a+b$ has a unique expression as an element of $A+B$ and $a+a$ has a unique expression as an element of $A+A$, then $A$ is a translate of $B$. We also give an explicit description of the various counterexamples which arise when neither condition holds.

We study conditions on automorphisms of Boolean algebras of the form $\left(\lambda \right)/{ℐ}_{\kappa }$ (where λ is an uncountable cardinal and ${ℐ}_{\kappa }$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $\left(2^{\kappa }\right)/{ℐ}_{\kappa ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M{A}_{\aleph ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial.

We present two ${\mathbb{P}}_{max}$ varations which create maximal models relative to certain counterexamples to Martin’s Axiom, in hope of separating certain classical statements which fall between MA and Suslin’s Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster’s forcing axiom ₃ fails. Of particular interest is the still open question...

We present principles for guessing clubs in the generalized club filter on ${}_{\kappa }\lambda$. These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ⁺-Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of ${\left[\lambda \right]}^{\aleph ₀}$.

A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality ℵ₁, and thus that there exist at least $2^{\aleph ₁}$ such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets of reals....

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

A function of two variables F(x,y) is universal if for every function G(x,y) there exist functions h(x) and k(y) such that G(x,y) = F(h(x),k(y)) for all x,y. Sierpiński showed that assuming the Continuum Hypothesis there exists a Borel function F(x,y) which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. Here we show that it is consistent that for each α with 2 ≤ α < ω₁ there...