Martin's maximum and the non-stationary ideal

We analyze the non-stationary ideal and the club filter at aleph_1 under MM

Dense ideals and cardinal arithmetic

From large cardinals we show the consistency of normal, fine, $\kappa$-complete $\lambda$-dense ideals on $\mathcal{P}_\kappa(\lambda)$ for successor $\kappa$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman

Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

μ\mu-Clubs of Pκ(λ)P_\kappa (\lambda) : Paradise in heaven

Let $\mu < \kappa < \lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u (\kappa, \lambda)$ is regular, where $u (\kappa, \lambda)$ denotes the least size of a cofinal subset in $(P_\kappa (\lambda), \subseteq)$, and cf$(\lambda) \not= \mu$, then (a) the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa (u (\kappa, \lambda))$ are isomorphic, and (b) the ideal dual to the $\mu$-club filter on $P_\kappa (\lambda)$ (and hence the restriction of the nonstationary ideal on $P_\kappa (\lambda)$ to sets of uniform cofinality $\mu$) is not $I_{\kappa, \lambda}$-$\frak{b}_{u (\kappa, \lambda)}$-saturated

μ\mu-Clubs OF Pκ(λ)P_\kappa (\lambda) : Paradise on earth

If $V = L$, and $\mu$, $\kappa$ and $\lambda$ are three infinite cardinals with $\mu = {\rm cf} (\mu) < \kappa = {\rm cf}(\kappa) \leq \lambda$, then, as shown in \cite{Heaven}, the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa (\lambda^{< \kappa})$ are isomorphic if and only if ${\rm cf} (\lambda) \not= \mu$. Now in $L$, $\lambda^{< \kappa}$ equals $u (\kappa, \lambda)$ (the least size of a cofinal subset in $(P_\kappa (\lambda), \subseteq)$) equals $\lambda$ if ${\rm cf} (\lambda) \geq \kappa$, and $\lambda^+$ otherwise. We show that, in ZFC, there are many triples $(\mu, \kappa, \lambda)$ for which ($u (\kappa, \lambda) > \lambda$ and) the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa (u (\kappa, \lambda))$ are isomorphic