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This principle is inconsistent: Let $T\subseteq\omega_1$ be stationary such that $T$ does not contain a club (such a set exists e.g. by Solovay splitting which is certainly overkill). Let for an arbit …

Suppose that $\kappa$ has the property that for every family $A\subseteq\omega^{\omega}$, if $|A|<\kappa$, then there exists some $g\in\omega^{\omega}$ such that for any $f\in A, \exists^{\infty}n\;f( …

First of all, it is clear that the given Partition $\{A_n\;|\;n\in\omega\}$ satisfies $A_n\notin D$ for any n, since $\omega\smallsetminus A_n\in F\subseteq D$ for any n (since $(\omega\smallsetminus …

Let $\kappa$ and $\lambda$ be cardinals. A thin $(\kappa,\lambda)$-list is a function $L:[\lambda]^{<\kappa}\longrightarrow [\lambda]^{<\kappa}$ such that for all $x\in[\lambda]^{<\kappa}$, $L(x)\subs …

The completeness game $G_{\gamma}(P)$ for a partial order $P$ has players COM and INC play alternatingly and descendingly elements of $P$ with player INC playing first and player COM playing at limit …