The combinatorial essence of supercompactness (Q450963)

In [Ann. Math. Logic 5, 165--198 (1973; Zbl 0262.02062)] \textit{T.~J. Jech} introduced the notion of \((\kappa,\lambda)\)-mess (called \(P_\kappa\lambda\)-list in this paper) as a generalization of \(\kappa\)-tree and used it to characterize strongly compact cardinals. In [Proc. Am. Math. Soc. 42, 279--285 (1974; Zbl 0279.02050)] \textit{M.~Magidor} extended Jech's result to supercompactness. Based on these characterizations, the author of the paper under review introduces the following combinatorial principles: \textbf{TP}\((\kappa,\lambda)\), \textbf{SP}\((\kappa,\lambda)\), \textbf{ITP}\((\kappa,\lambda)\) and \textbf{ISP}\((\kappa,\lambda)\). He also defines and investigates the canonical ideals associated to the latter two principles.NEWLINENEWLINEThese principles are consistency-wise lower than the existence of a supercompact cardinal (see Corollary 5.6). In particular, the author shows that the principle \textbf{ITP}\((\kappa,\lambda)\) implies the failure of a variant of Schimmerling's two-cardinal square principle (Theorem 4.2), giving lower bounds on its consistency strength. By results from inner model theory, the consistency of \textbf{ITP}\((\kappa,\lambda^+)\) for suitable \(\kappa\), \(\lambda \) (Theorem 5.10 and its corollaries) implies that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals.NEWLINENEWLINELet us say a regular uncountable cardinal \(\kappa\) has the \textit{super tree property} if \textbf{ITP}\((\kappa,\lambda)\) holds for all \(\lambda\geq \kappa\). The author proves that for every \(n\geq 2\) there is a model of the super tree property for \(\aleph_n\), assuming the existence of one supercompact cardinal (see Corollary 5.6 for the case \(n=2\)). Later, \textit{L.~Fontanella} [J. Symb. Log. 78, No. 1, 317--333 (2013; Zbl 1279.03070)] and Unger independently obtained a model in which every \(\aleph_n\), \(n\geq 2\), satisfies the super tree property, assuming the existence of infinitely many supercompact cardinals.NEWLINENEWLINEAt the end of the paper, the author also mentions his joint work with \textit{M.~Viale} [Adv. Math. 228, No. 5, 2672--2687 (2011; Zbl 1251.03059)], which includes an interesting application of these principles in constructing models of \textbf{PFA} from large cardinal assumptions.NEWLINENEWLINE The results in this paper are from the author's PhD dissertation [Subtle and ineffable tree properties. PhD thesis. München: Ludwig {M}aximilians {U}niversität (2010), \url{http://edoc.ub.uni-muenchen.de/11438}].