Marcin Michalski

The two‐dimensional version of the classical Mycielski theorem says that for every comeager or conull set there exists a perfect set such that. We consider a strengthening of this theorem by replacing a perfect square with a rectangle, where A and B are bodies of some types of trees with. In particular, we show that for every comeager Gδ set there exist a Miller tree and a uniformly perfect tree such that and that cannot be a Miller tree. In the case of measure we show that for every subset F of of full measure there exists a uniformly perfect tree such that and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions from which we derive nonstandard proofs of Mycielski‐like theorems via the Shoenfield Absoluteness Theorem.

A \(\sigma \) -ideal \(\mathcal {I}\) on a Polish group \((X,+)\) has the Smital Property if for every dense set _D_ and a Borel \(\mathcal {I}\) -positive set _B_ the algebraic sum \(D+B\) is a complement of a set from \(\mathcal {I}\). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are \(\mathfrak {c}\) many maximal invariant \(\sigma \) -ideals with Borel bases on the Cantor space \(2^\omega \).

The paper attempts to present a diachronic view of some syntactic constructions with the Arabic elative. The point of departure is its early stage in Classical Arabic. Since then, it has undergone substantial development, resulting in modern syntactic uses unknown to traditional Arabic grammarians. Of special interest are the historical trajectories of and semantic relations between the three constructions conveying the meaning of the superlative that are in current use in Modern Written Arabic: elative + indefinite singular noun in the genitive, elative + definite plural noun in the genitive, and elative used as an attribute of a definite noun – all of them translatable as ‘the largest city’, but different from each other in certain respects. A tendency to create symmetry can be observed in the system of the syntax of the elative used in adjectival attribution on the one hand and the manner of expressing a gradation comparative-superlative by means of the same syntactic construction differing only in definiteness, on the other hand. In addition, a construction only marginally attested in CA, viz. elatives following the pattern C1uC2C3ā + definite plural noun in the genitive, turns out to be used widely in MWA. Its plural counterpart, C1uC2C3āayāt, is an innovation whose expansive usage, especially in the journalistic style, can be attributed to the tendency of disambiguation.

In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, $cl_0$, $h_0,$ and $ch_0$. We show that there exists a subset of the Baire space $\omega ^\omega,$ which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of ${\mathbb {T}}$ -Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees ${\mathbb {T}}$. We also obtain a result on ${\mathcal {I}}$ -Luzin sets, namely, we prove that if ${\mathfrak {c}}$ is a regular cardinal, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$, $l_0,$ and $cl_0$.