Miroslav Repick´y Saharon Shelah on Tree Ideals

Let l and m be the ideals associated with Laver and Miller forcing, respectively. We show that add(l) < cov(l) and add(m) < cov(m) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal ≤ h. Introduction and Notation In this paper we investigate the ideals connected with the classical tree forcings introduced by Laver [La] and Miller [Mi]. Laver forcing L is the set of all trees p on ω such that p has a stem and whenever s ∈ p extends stem(p) then Succp(s) := {n : sˆn ∈ p} is infinite. Miller forcing M is the set of all trees p on ω such that p has a stem and for every s ∈ p there is t ∈ p extending s such that Succp(t) is infinite. The set of all these splitting nodes in p we denote by Split(p). For any t ∈ Split(p), Splitp(t) is the set of all minimal (with respect to extension) members of Split(p) which properly extend t. For both L and M the order is inclusion. The Laver ideal l is the set of all X ⊆ ω with the property that for every p ∈ L there is q ∈ L extending p such that X ∩ [q] = ∅. Here [q] denotes the set of all branches of q. The Miller ideal m is defined analogously, using conditions in M instead of L. By a fusion argument one easily shows that l and m are σ-ideals. The additivity (add) of any ideal is defined as the minimal cardinality of a family of sets belonging to the ideal whose union does not. The covering number (cov) is defined as the least cardinality of a family of sets from the ideal whose union is the whole set on which the ideal is defined – ω in our case. Clearly ω1 ≤ add(l ) ≤ cov(l) ≤ c and ω1 ≤ add(m) ≤ cov(m) ≤ c hold. The main result in this paper says that there is a model of ZFC where add(l) < cov(l) and add(m) < cov(m) hold. The motivation was that by a result of Plewik [Pl] it was known that the additivity and the covering number of the ideal connected with Mathias forcing are the same and they are equal to the cardinal invariant h – the least cardinality of a family of maximal antichains of P(ω)/fin without a common refinement. On the other hand, in [JuMiSh] it was shown that add(s) < cov(s) is consistent, where 1 Supported by DFG grant Ko 490/7-1, and by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany) 2 Publication 487. 3 Supported by the Basic Reasearch Foundation of the Israel Academy of Sciences and the Schweizer Nationalfonds