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 study and classify topologically invariant σ-ideals with an analytic base on Euclidean spaces, and evaluate the cardinal characteristics of such ideals.

We formulate variants of the cardinals f, p and t in terms of families of measurable functions, in order to examine the effect upon these cardinals of adding one random real.

We compute transitive cardinal coefficients of ideals on generalized Cantor spaces. In particular, we show that there exists a null set $A\subseteq 2^{\omega ₁}$ such that for every null set $B\subseteq 2^{\omega ₁}$ we can find $x\in 2^{\omega ₁}$ such that A ∪ (A+x) cannot be covered by any translation of B.

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....