Mathias absoluteness and the Ramsey property

In this article we give a forcing characterization for the Ramsey property of -Sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of real

Zastosowania technik macierzy bazowych w zakresie forcingowych częściowych porządków

The cr-idcal (f°) is associatcd with the Silvcr forcing, see [8]. Also, it constitutes the family of all completely doughnut nuli sets, see [19]. We introduce segment topologies to state some resemblances of (v°) to the family of Ramsey nuli sets. To describe add(v°) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v°) = add(v°) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v°) = uji implies that (w0) has the ideał type c)