ℛmax variations for separating club guessing principles

Abstract

In his book on ℛ_max [7], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω₁. In this paper we employ one of the techniques from this book to produce ℛ_max variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [4] while studying games of length ω₁. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω₁. In this paper we give an alternate proof of the second of these results, using Woodin's ℛ_max technology, showing that a strengthening of (+) does not imply a weakening of club guessing known as the Interval Hitting Principle. The main technique in this paper, in addition to the standard ℛ_max machinery, is the use of condensation principles to build suitable iterations.