Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle.
Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of an higher Souslin tree.
We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper.
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Citation information:
A. Rinot, On guessing generalized clubs at the successors of regulars, Ann. Pure Appl. Logic, 162(7): 566-577, 2011.
Question 2 has been answered (in the negative) by Paul Larson. See here.
An affirmative answer to Question 1 follows from recent work of Aspero in his paper “The consistency of a club-guessing failure at the successor of a regular cardinal”.