$^\kappa\kappa$ in light of the Tukey ordering(Forcing and Infinitary Combinatorics)
Sakaé Fuchino, Masayuki Karato, Hiroshi Sakai, Toshimichi Usuba · Institutional Repositories DataBase (IRDB) · 2006
For a regular $/\sigma$ , we show that, under $\kappa^{t\mathrm{t}^{+}$ as well as in any tt-c.c.extension of a model of $\kappa^{\kappa^{+}$ , there is no subset $F$ of $\hslash\kappa$ of cardinality $\kappa^{+}$ such that $F$ has less than $\kappa$ many elements below every $g\in$ $\kappa\iota \mathrm{s}$ with respect to the partial ordering $\leq$ on $\kappa/\sigma$ by coordinatewise comparison.By Lemma 3.5 in Karato [7], the non-existence of such 7 is equivalent to $\langle \mathcal{P}_{\kappa}\kappa^{+}, \subseteq\rangle ot\leq(^{\kappa}\kappa,$ $\leq\rangle$ in the Tukey ordering.Todorcevic pointed out that this condition is actually equivalent to what is called Galvin's proposition for $\kappa$ inAbraham and Shelah [1].Thus our arguments provide an alternative proof of Galvin's proposition, By this equivalence, a result in Abraham and Shelah [1] reads e.g. that $\langle \mathcal{P}_{\omega_{n}}\omega_{m}, \subseteq\rangle\leq\langle\langle^{\omega_{n}}\omega_{n}, \leq\rangle$ is consistent for any $1\leq n<m<\omega$ .We also show that { $P_{\hslash}\lambda,$ $\subseteq\rangle ot\leq\langle^{\kappa}\kappa,$ $\leq$ ) holds if A has a certain large cardinal property.