F U N D a M E N T a Mathematicae Strolling through Paradise

With each of the classical tree-like forcings adjoining a new real, one can associate a σ-ideal on the reals in a natural way. For example, the ideal s0 of Marczewski null sets corresponds to Sacks forcing S, while the ideal r0 of nowhere Ramsey sets corresponds to Mathias forcing R. We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal P2, and start an investigation of the covering numbers of these ideals. Introduction. In 1935, E. Marczewski [Mar] introduced on the reals the σ-ideal s0, consisting of sets X ⊆ 2 so that for all perfect trees T ⊆ 2 there is a perfect subtree S ⊆ T with [S] ∩X = ∅, where [S] := {f ∈ ω : ∀n (f1n ∈ S)} denotes the set of branches through S (see [JMS], [Mi2], [Ve] and others for recent results on s0). Similarly a set X ⊆ 2 is called s-measurable iff for all perfect trees T there is a perfect subtree S ⊆ T with either [S]∩X = ∅ or [S] ⊆ X. Once forcing was born, the algebra of smeasurable sets modulo s0-sets turned out to be of great interest; it was first studied by G. Sacks [Sa], and henceforth became known as Sacks (or perfect set) forcing S. Since then, many Sacks-like partial orders have been investigated (e.g., Mathias forcing R, Laver forcing L, Miller forcingM etc.—see §1 for the definitions), and it is natural to ask how the corresponding σ-ideals (i.e. r0, `0, m0, respectively) look like. We note that the ideal r0 of Ramsey null (or nowhere Ramsey) sets was first considered by Galvin and Prikry [GP], and has found much attention over the years (see [AFP], [Br], [Co], [Ma2], [Pl] and others), while `0 and m0 were looked at only recently in work of Goldstern, Johnson, Repický, Shelah and Spinas (see [GJS] and [GRSS]). 1991 Mathematics Subject Classification: 03E05, 03E35, 03E40.