Two-cardinal diamond principles and saturation of the nonstationary ideal - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:06:14Z http://mathoverflow.net/feeds/question/77074 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77074/two-cardinal-diamond-principles-and-saturation-of-the-nonstationary-ideal Two-cardinal diamond principles and saturation of the nonstationary ideal Trevor Wilson 2011-10-03T20:16:29Z 2012-08-12T00:22:02Z <p>In the paper "Stationary reflection and the club filter", the author Masahiro Shioya says that the club filter on $P_{\omega_1}(\lambda)$ cannot be $2^\lambda$-saturated for $\lambda > \omega_1$, citing Shelah's book "Nonstructure Theory" (in preparation). I have three questions:</p> <p>1) Is there a published reference for this result?</p> <p>2) Does the theorem apply to $P_{\omega_1}(\lambda) | S$ for an arbitrary stationary set $S$?</p> <p>3) Does the proof go through a two-cardinal diamond principle? I.e., did Shelah prove (in ZFC) that $\lozenge_{\omega_1,\lambda}$ holds for $\lambda > \omega_1$? What about $\lozenge_{\omega_1,\lambda}(S)$ for arbitrary stationary $S$?</p> <p>I am particularly interested in the case $\lambda = 2^{\omega} = \omega_2$. In this case $\lozenge_{\omega_1,\lambda}(S)$ was proved by Donder and Matet in the paper "Two cardinal versions of diamond" for stationary sets $S$ of the form $\lbrace a \in P_{\omega_1}(\lambda) : \sup a \in B\rbrace$ where $B \subset \lambda$ is a stationary set consisting of points of cofinality $\omega$. Does this hold for arbitrary stationary $S$?</p>