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This is probably well known but I'll appreciate pointers to references: Is there any model where for a singular cardinal $\kappa$ of cofinality $\omega$, Very Weak Square holds at $\kappa$ but every …

Suppose $\kappa$ carries an $\omega_1$-saturated $\kappa$-complete ideal $I$, given $M\prec (V_{\kappa+2},\in , <)$ ($<$ well orders $V_{\kappa+2}$) of size $<\kappa$ containing $I$, we show how to fi …

The following may give a hint: Suppose $U$ is a uniform ultrafilter on $\omega$ and $W$ is an ultrafilter on $\kappa$ such that $W$ commutes with $U$, then $W$ is countably complete. Otherwise, there …

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, …

In Woodin's book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal" Remark 2.55 (5), it states SRP by Todorcevic (defined below) is consistent with the existence of a Suslin tree …

Aside from the well-known characterization of weakly compact cardinals in terms of the usual partition calculus, I've been wondering if there are other characterizations that are variants of the typic …

Rado's conjecture holds in Mitchell's model (of course, start with a strongly compact instead of a weakly compact) granted the following: If $T$ if a non-special tree of height $\omega_1$, then $T$ re …

Rado's conjecture (one of many equivalent formulations) states: any non-special tree has a non-special subtree of cardinality $\aleph_1$. "Special" means a tree can be decomposed into countably many …