Abstract We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa $ , the existence of a strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is a theorem of $\textsf{ZFC}$ . Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is independent of $\textsf{ZFC}$ . We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $\kappa $ -Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is equivalent to a certain weak indexed square principle $\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa , \theta )$ . We conclude the paper with an application to the failure of the infinite productivity of $\kappa $ -stationarily layered posets, answering a question of Cox.

Abstract Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at ${\kappa ^{ + + }}$ , assuming that $\kappa = {\kappa ^{ < \kappa }}$ and there is a weakly compact cardinal aboveκ. If in additionκis supercompact then we can forceκto be ${\aleph _\omega }$ in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$ -Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$ .

Abstract 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 in L for every infinite cardinal. As an application, we prove that the following two hold in L: 1. For every infinite regular cardinal λ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin; 2. For every infinite cardinal λ, there exists a respecting λ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed in L.

Abstract Letλdenote a singular cardinal. Zeman, improving a previous result of Shelah, proved that together with 2λ=λ+implies ⋄Sfor everyS ⊆ λ+that reflects stationarily often. In this paper, for a setS ⊆ λ+, a normal subideal of the weak approachability ideal is introduced, and denoted byI[S; λ]. We say that the ideal isfatif it contains a stationary set. It is proved: 1. ifI[S; λ] is fat, then NSλ+ ∣Sis non-saturated; 2. ifI[S; λ] is fat and 2λ= λ+, then ⋄Sholds; 3. implies thatI[S; λ] is fat for everyS⊆λ+that reflects stationarily often; 4. it is relatively consistent with the existence of a supercompact cardinal that fails, whileI[S; λ] is fat for every stationaryS ⊆ λ+that reflects stationarily often. The stronger principle is studied as well.