# Sh:685

• Džamonja, M., & Shelah, S. (2000). On versions of \clubsuit on cardinals larger than \aleph_1. Math. Japon., 51(1), 53–61.
• 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 [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 Džamonja-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 Džamonja-Shelah result.
Bib entry
@article{Sh:685,
author = {D{\v{z}}amonja, Mirna and Shelah, Saharon},
title = {{On versions of $\clubsuit$ on cardinals larger than $\aleph_1$}},
journal = {Math. Japon.},
fjournal = {Mathematica Japonica},
volume = {51},
number = {1},
year = {2000},
pages = {53--61},
issn = {0025-5513},
mrnumber = {1739051},
mrclass = {03E05 (03E35)},
note = {\href{https://arxiv.org/abs/math/9911228}{arXiv: math/9911228}},
arxiv_number = {math/9911228}
}