The special Aronszajn tree property

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal [Formula: see text], there are [Formula: see text]-Aronszajn trees, and all such trees are special.

SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

SQUARE WITH BUILT-IN DIAMOND-PLUS

We formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds inLfor every infinite cardinal.As an application, we prove that the following two hold inL:1.For every infinite regular cardinalλ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;2.For every infinite cardinalλ, there exists arespectingλ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed inL.

CHAIN CONDITIONS OF PRODUCTS, AND WEAKLY COMPACT CARDINALS

The history of productivity of theκ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal$\kappa > \aleph _1 {\rm{,}}$the principle □(k) is equivalent to the existence of a certain strong coloring$c\,:\,[k]^2 \, \to $kfor which the family of fibers${\cal T}\left( c \right)$is a nonspecialκ-Aronszajn tree.The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if theκ-chain condition is productive for a given regular cardinal$\kappa > \aleph _1 {\rm{,}}$thenκis weakly compact in some inner model of ZFC. This provides a partial converse to the fact that ifκis a weakly compact cardinal, then theκ-chain condition is productive.

Creatures on ω1 and weak diamonds

We specialise Aronszajn trees by an ωω-bounding forcing that adds reals. We work with creature forcings on uncountable spaces.As an application of these notions of forcing, we answer a question of Moore, Hrušák and Džamonja whether implies the existence of a Souslin tree in a negative way by showing that “ and every Aronszajn tree is special” is consistent relative to ZFC.