In [Two examples of Borel partially ordered sets with the countable chain condition, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1125–1128], Todorcevic introduced a ccc forcing which is Borel definable in a separable metric space. In [On Todorcevic orderings, Fund. Math., to appear], Balcar, Pazák and Thümmel applied it to more general topological spaces and called such forcings Todorcevic orderings. There they analyze Todorcevic orderings quite deeply. A significant remark is that Thümmel solved...

We investigate $\left|\right|{\chi }_{𝔸},2\left|\right|$, the minimum cardinality of a subset of $2^{\omega }$ that cannot be made convergent by multiplication with a single matrix taken from $𝔸$, for different sets $𝔸$ of Toeplitz matrices, and show that for some sets $𝔸$ it coincides with the splitting number. We show that there is no Galois-Tukey connection from the chaos relation on the diagonal matrices to the chaos relation on the Toeplitz matrices with the identity on $2^{\omega }$ as first component. With Suslin c.c.c. forcing we show that $\left|\right|{\chi }_{𝕄},2\left|\right|$ < $\bullet$ is consistent...

We describe a totally proper notion of forcing that can be used to shoot uncountable free sequences through certain countably compact non-compact spaces. This is almost (but not quite!) enough to produce a model of ZFC + CH in which countably tight compact spaces are sequential-we still do not know if the notion of forcing described in the paper can be iterated without adding reals.

We deal with two cardinal invariants and give conditions on their equality using Shelah's pcf theory.

We prove:  Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a κ-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph $G_0$ into G.  Theorem 2. In the Solovay model, every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above.

A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some $\sigma$-ideal, being (completely) nonmeasurable with respect to different $\sigma$-ideals, being a $\kappa$-covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations...

We characterize for which ultrafilters on $\omega$ is the ultrafilter extension of the asymptotic density on natural numbers $\sigma$-additive on the quotient boolean algebra $𝒫\left(\omega \right)/d_{𝒰}$ or satisfies similar additive condition on $𝒫\left(\omega \right)/\text{fin}$. These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C., A Note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320] under the name $AP$(null) and $AP$(*). We also present a characterization of a $P$- and semiselective ultrafilters...

We formulate a Covering Property Axiom $CP{A}_{cube}^{game}$, which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while $CP{A}_{cube}^{game}$ implies that every universally null has cardinality less than = ω₂. We also show that $CP{A}_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null compact sets....

Natural weakenings of uniformizability of a ladder system on ω₁ are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The...

We investigate the question of whether or not an amenable subgroup of the permutation group on $\mathbb{N}$ can have a unique invariant mean on its action. We extend the work of M. Foreman (1994) and show that in the Cohen model such an amenable group with a unique invariant mean must fail to have slow growth rate and a certain weakened solvability condition.

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...

We show that universal indestructibility for both strong compactness and supercompactness is consistent with the existence of two strongly compact cardinals. This is in contrast to the fact that if κ is supercompact and universal indestructibility for either strong compactness or supercompactness holds, then no cardinal λ > κ is measurable.

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 consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most ${\aleph }_2$, “not” if the set is allowed to be of size ${\left(2^{2^{{\aleph }_0}}\right)}^{+}$. It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.