On versions of clubsuit on cardinals larger than aleph_1

Abstract

We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambda^+ and lambda is a strong limit singular, then the corresponding guessing holds. Both results are also connected to an earlier result of Dzamonja and Shelah in which they showed that a certain version of clubsuit holds at a successor of singular just in ZFC. The first result here shows that a result of math.LO/9601219 can to a certain extent be extended to the successor of a regular. The negative result here gives limitations to the extent to which one can hope to extend this result.