Universal forcing notions and ideals

The main result of this paper is a partial answer to [6, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ–ideals determined by those universality parameters. One of the most striking differences between measure and category was discovered in Shelah [8] where it was proved that the Lebesgue measurability of Σ 1 3 sets implies ω 1 is inaccessible in L, while one can construct (in ZFC) a forcing notion P such that V P |= " projective subsets of R have the Baire property ". For the latter result one builds a homogeneous ccc forcing notion adding a lot of Cohen reals. Homogeneity is obtained by multiple use of amalgamation (see [4] for a full explanation of how this works), the Cohen reals come from compositions with the Universal Meager forcing notion UM or with the Hechler forcing notion D. The main point of that construction was isolating a strong version of ccc, so called sweetness, which is preserved in amalgamations. Later, Stern [10] introduced a weaker property, topological sweetness, which is also preserved in amalgamations. Sweet (i.e., strong ccc) properties of forcing notions were further investigated in [6], where we introduced a new property called iterable sweetness (see [6, Definition 4.2.1]) and we proved the following two results. If P is a sweet ccc forcing notion (in the sense of [8, Definition 7.2]) in which any two compatible elements have a least upper bound, then P is iterably sweet. (2) (See [6, Theorem 4.2.4]) If P is a topologically sweet forcing notion (in the sens of Stern [10, Definition 1.2]) and Q ˜ is a P–name for an iterably sweet forcing, then the composition P * Q ˜ is topologically sweet. In [6, §2.3] we introduced a scheme of building forcing notions from so called universality parameters (see 1.2 later). We proved that typically they are sweet (see [6, Proposition 4.2.5]) and in natural cases also iterably sweet. So the question arose if the use of those forcing notions in iterations gives us something really new. Specifically, we asked: The first author thanks the Hebrew University of Jerusalem for support during his visit to Jerusalem in Spring'2003. He also thanks his wife, Ma lgorzata Jankowiak–Ros lanowska for supporting him when he was working on this …