Hechler and Laver Trees

TL;DR

This paper explores the properties of Hechler and Laver trees in relation to analytic sets, proving their connection to Ramsey theory and analyzing their behavior under different set-theoretic assumptions.

Contribution

It establishes a link between Hechler and Laver trees and analytic sets, proving a Ramsey property and examining the impact of set-theoretic axioms.

Findings

Every analytic set is either disjoint from Hechler tree branches or contains Laver tree branches.

Under G"odel's constructible universe, the result fails for co-analytic sets.

Under Martin's axiom, the result holds for 3^1_2 sets.

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