Subtlety and partition relations (Q2793908)

This paper investigates different variants of the notions of subtle and faint cardinals and sets. Subtle and faint cardinals are large cardinal notions that were defined by Jensen and Kunen in unpublished manuscripts as partial answser to the question of which large cardinals imply the diamond principle. Subtle and faint cardinals (as well as their stronger version -- the ineffable cardinals), play a major role in the investigation of large cardinals below \(0^\#\). Using the miniaturization terminology from [\textit{M. Džamonja} and \textit{J. D. Hamkins}, Ann. Pure Appl. Logic 144, No. 1--3, 83--95 (2006; Zbl 1110.03032)], if weakly compact cardinals are miniature versions of measurable cardinals then subtle cardinals are miniature versions of Woodin cardinals.NEWLINENEWLINEFor a cardinal \(\kappa\), let \(I\) be the ideal of all sets \(X \subseteq \kappa\) such that there is a sequence \(\langle a_\alpha \mid \alpha \in X\rangle\), with the property that for all \(\alpha\), \(a_\alpha \subseteq \alpha\) and for every \(\alpha < \beta\) in \(X\), \(a_\alpha \neq a_\beta \cap \alpha\). We define the ideals \(\mathrm{NF}_\kappa = I + J_{bd}\) (where \(J_{bd}\) is the ideal of bounded subsets of \(\kappa\)) and \(\mathrm{NS}u_\kappa = I + \mathrm{NS}_\kappa\) (where \(\mathrm{NS}_\kappa\) is the ideal of non-stationary subsets of \(\kappa\)). \(\kappa\) is faint if \(\kappa\notin \mathrm{NF}_\kappa\). \(\kappa\) is subtle if \(\kappa \notin \mathrm{NS}u_\kappa\).NEWLINENEWLINEThe author studies the behavior of \(\mathrm{NF}_\kappa\) and \(\mathrm{NS}u_\kappa\) and shows that \(\mathrm{NF}_\kappa\) is determined from \(\mathrm{NS}u_\alpha\) for \(\alpha \leq \kappa\). In order to get this surprising result, the author uses a generalization of the partition relation characterization, due to Baumgartner, of subtle subsets of \(\kappa\). Baumgartner showed that a set \(X\subseteq \kappa\) is not in \(\mathrm{NS}u_\kappa\) iff every regressive coloring of pairs of \(X \cap C\) (where \(C\) is a club at \(\kappa\)) has a homogeneous sets of size \(3\) (see [\textit{J. E. Baumgartner}, in: Logic, Found. Math., Comput. Theory; Proc. 5th int. Congr., London/Ontario 1975, Part 1, 87--106 (1977; Zbl 0373.04002); \textit{H. M. Friedman}, Ann. Pure Appl. Logic 107, No. 1--3, 1--34 (2001; Zbl 0966.03048)]). In this paper, the author generalizes this result to \(\mathrm{NF}_\kappa\) and uses it to obtain some powerful results about the ideal \(\mathrm{NF}_\kappa\).NEWLINENEWLINEThe notions of faintness and subtlety of sets were generalized to the context of \(\mathcal{P}_\kappa \gamma\) by \textit{T. K. Menas} [J. Symb. Log. 41, 225--234 (1976; Zbl 0331.02045)]. The relation ``\(<\)'' in the definition of faintness and subtlety has two natural versions in \(\mathcal{P}_\kappa\gamma\): inclusion and strong inclusion. Thus, there are (at least) two natural generalizations of subtlety and faintness in \(\mathcal{P}_\kappa\gamma\). The author shows that it is consistent relative to a measurable that \(\mathcal{P}_\kappa \gamma\) can be subtle with respect to inclusion but not subtle with respect to strong inclusion.NEWLINENEWLINEThe author shows that many properties of the ideals \(\mathrm{NF}_\kappa, \mathrm{NS}u_\kappa\) generalize to their \(\mathcal{P}_\kappa\gamma\) versions. In particular, there is a natural partition relation characterization for faintness and subtlety of subsets of \(\mathcal{P}_\kappa \gamma\).NEWLINENEWLINEMenas [loc. cit.] showed that if \(\kappa\) is subtle then for all \(\gamma\geq\kappa\), \(\mathcal{P}_\kappa \gamma\) is subtle. The author shows that the other direction can fail. Specifically, if \(\gamma\) is measurable then there is \(\kappa < \gamma\) which is faint but not subtle such that \(\mathcal{P}_\kappa \gamma\) is subtle. The author investigates this situation and shows that its consistency strength is higher than the consistency strength of subtle cardinal (it is somewhere between \(0^\#\) and a measurable cardinal).