On Tree Ideals

Let Io and m° be the ideals associated with Laver and Miller forcing, respectively. We show that add(/°) < cov(/°) and add(m°) < cov(m°) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal < f). 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 <(0co such that p has a stem and whenever s £ p extends stem(p) then Succp(s) := {n : s~n £ p} is infinite. Miller forcing M is the set of all trees p on <wco such that p has a stem and for every s £ p there is t £ p extending s such that Succp(t) is infinite. We denote the set of all these splitting nodes in p by Split(p). For any t g 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 Io is the set of all X C aco with the property that for every p £ L there is q £ L extending p such that X n [q] 0. 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 Io and m° are er-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— "co in our case. Clearly cox < add(/°) < cov(/°) < c and cox < add(m°) < cov(w°) < c hold. Received by the editors January 22, 1993 and, in revised form, August 9, 1993. 1991 Mathematics Subject Classification. Primary 03E35, 03E50, 54A25. The first and third authors were 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). Publication 487. The second author was supported by the Basic Research Foundation of the Israel Academy of Sciences and by grant GA SAV 365 of the Slovak Academy of Sciences. The fourth author was supported by the Basic Research Foundation of the Israel Academy of Sciences and the Schweizer Nationalfonds. ©1995 American Mathematical Society 0002-9939/95 $1.00+ $.25 per page