Hechler and Laver Trees 1 Hechler and Laver Trees

A Laver tree is a tree in which each node splits infinitely often. A Hechler tree is a tree in which each node splits cofinitely often. We show that every analytic set is either disjoint from the branches of a Heckler tree or contains the branches of a Laver tree. As a corollary we deduce Silver Theorem that all analytic sets are Ramsey. We show that in Godel’s constructible universe that our result is false for co-analytic sets (equivalently it fails for analytic sets if we switch Hechler and Laver). We show that under Martin’s axiom that our result holds for Σ12 sets. Finally we define two games related to this property. Definition 1 A subtree H ⊆ ω<ω is Hechler iff ∀s ∈ H ∀∞n sn ∈ H. A subtree L ⊆ ω<ω is Laver iff ∀s ∈ L ∃∞n sn ∈ L. These definitions are motivated by well-known forcing notions of Laver [4] and Hechler [3]. In the classical Hechler forcing the cofinite sets on the nth level of the tree would all be the same. Definition 2 For any subtree T ⊆ ω<ω define [T] = {x ∈ ωω: ∀n x n ∈ T} Theorem 3 For any Σ11 set A ⊆ ωω either there exists a Hechler tree H with [H] ∩ A = ∅ or there exists a Laver tree L with [L] ⊆ A. 1These results were obtained in December 2002 while visiting the Fields Institute at the University of Toronto. Thanks to its director Juris Steprans and his staff for their hospitality and also for the discussions we had with Steprans at that time. It was presente