PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 4, Pages 1183–1187 S 0002-9939(01)06174-3 Article electronically published on October 1, 2001 STRONGLY MEAGER SETS OF REAL NUMBERS AND TREE FORCING NOTIONS ANDRZEJ NOWIK AND TOMASZ WEISS (Communicated by Carl G. Jockusch, Jr.) Abstract. We show that every strongly meager set has the l 0 - and the m 0 - property. In [NW] it was proven that every strongly meager subset of 2 ω is a completely Ramsey null (CR 0 ) set. In this paper we show that an analogous result holds for Laver and Miller notions of forcing, i.e., every strongly meager set has both the l 0 - and the m 0 -property. Notice that the three classes of subsets of 2 ω mentioned above are not related as far as inclusion is concerned. This fact is due to J. Brendle (see [B]). Our terminology is standard and can be found in [B] and [NW]. If T is a tree on ω <ω↑ , the set of all increasing sequences of natural numbers, then the stem(T ) of T is the unique s ∈ T (if such s exists) with ∀ t∈T s ⊆ t ∨ t ⊆ s and |{n ∈ ω : s⌢ 〈n〉∈ T }| ≥ 2. Given t ∈ T , we define succ T (t)= {n ∈ ω : t⌢ 〈n〉∈ T } and we put split(T )= {s ∈ T : |succ T (s)|≥ 2}. For s ∈ T , Succ T (s)= {t ∈ ω <ω↑ : s⌢t ∈ split(T ) ∧∀ t ′ ∈split(T ) s ⊆ t ′ ⊆ s⌢t ⇒ (t ′ = s ∨ t ′ = s⌢t)}. Finally, [T ]= {x ∈ ω ω↑ : ∀ n∈ω x | ` n ∈ T }. We recall that a tree T ⊆ ω <ω↑ is said to be a Laver tree iff for every s ∈ T with stem(T ) ⊆ s, succ T (s) is infinite. A tree T ⊆ ω <ω↑ is a superperfect tree iff for any given s ∈ T , there is t ⊇ s such that succ T (t) is infinite. Let [ω] ω be the set of all infinite subsets of ω. It is clear that an x ∈ ω ω↑ can be identified with an element of [ω] ω and vice versa. Thus, depending on the context, ω ω↑ is often conflated with [ω] ω or with the set {x : x ∈ 2 ω and ∃ ∞ n x(n)=1}. Throughout this paper we assume that the measure used on 2 ω is the product measure. By “+” we mean the usual modulo 2 coordinatewise addition in 2 ω and for sets A, B ⊆ 2 ω we define A + B = {a + b : a ∈ A, b ∈ B}. Let us recall that X ⊆ 2 ω is strongly meager iff for every measure zero set A ⊆ 2 ω , there is t ∈ 2 ω , so that (X + t) ∩ A = ∅. Definition 1. We say that X ⊆ [ω] ω is an l 0 - set iff for every Laver tree T , there exists a Laver tree S ⊆ T such that {ran(x): x ∈ [S]}∩ X = ∅. Definition 2. A set X ⊆ [ω] ω is called an m 0 - set iff for every superperfect tree T , there is a superperfect tree S ⊆ T such that {ran(x): x ∈ [S]}∩ X = ∅. Received by the editors July 7, 2000 and, in revised form, October 2, 2000. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15. Key words and phrases. Strongly meager sets, Laver forcing, Miller forcing. The first author was partially supported by KBN grant 2 P03A 047 09. c 2001 American Mathematical Society 1183 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use