### On versions of $\clubsuit$ on cardinals larger than $\aleph_1$

by Dzamonja and Shelah. [DjSh:685]

Math Japonica, 2000

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
[Sh:667] 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 D{z}amonja-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 [DjSh:545] 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 the mentioned D{z}amonja-Shelah result.

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