PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 9, September 2007, Pages 2975–2982 S 0002-9939(07)08808-9 Article electronically published on May 9, 2007 SPECIAL SUBSETS OF THE REALS AND TREE FORCING NOTIONS MARCIN KYSIAK, ANDRZEJ NOWIK, AND TOMASZ WEISS (Communicated by Julia Knight) Abstract. We study relationships between classes of special subsets of the reals (e.g. meager-additive sets, γ-sets, C ′′ -sets, λ-sets) and the ideals related to the forcing notions of Laver, Mathias, Miller and Silver. 1. Introduction This paper is meant to be a sequel of [5] and [11], where the relationships between various classes of small sets and the ideals (l 0 ) and (m 0 ) related to the forcing notions of Laver and Miller were studied. In the present paper, we add the Silver forcing to the list of the forcing notions under consideration. We begin with studying inclusions between the classes of small sets already investigated in [5], and we also consider the σ-ideal (v 0 ) related naturally to the forcing notion of Silver. Later we take into consideration more classes of special subsets of the real line and we study their relationships with all the three ideals (l 0 ), (m 0 ) and (v 0 ). We also mention the classical ideal (cr 0 ) of completely Ramsey-null sets related to the Mathias forcing, solving a problem of J. Brown concerning σ-sets. The definitions of Laver and Miller trees as well as the ideals (l 0 ), (m 0 ) and the classes of perfectly meager, strongly null and universally null sets are provided in [5]. We continue using all the notation and terminology from above. Here, we shall sometimes abuse terminology by conflating a tree T with the set [T ] of its branches. Let us briefly remind the reader that we treat the Baire space ω ω as a co- countable subset of the Cantor space 2 ω . This identification allows us to think about (sets of branches of) Laver and Miller trees as subsets of 2 ω . The embedding was described precisely in [5]; let us just say that first we identify ω ω with the space of strictly increasing sequences of natural numbers ω ↑ω (and the homeomorphism we use preserves Laver and Miller trees). Then a strictly increasing sequence is identified with the set of its values, which is an element of [ω] ω ⊆ 2 ω . Received by the editors March 13, 2006 and, in revised form, June 8, 2006. 2000 Mathematics Subject Classification. Primary 03E05, 03E35, 28E15, 54G99. Key words and phrases. γ-set, Rothberger’s property, meager-additive set, σ-set, Laver forcing, Miller forcing, Silver forcing, completely Ramsey-null set. A part of the research was made when the first author was visiting the Institute of Mathematics of the Polish Academy of Sciences. The second author was partially supported by grant BW/5100-5-0201-6. c 2007 American Mathematical Society 2975 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use