Hechler and Laver Trees
Arnold W. Miller
Abstract
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 Sigma^1_2
sets. Finally we define two games related to this property.
Latex2e 8 pages
Latest version at
http://www.math.wisc.edu/~miller/res/index.html