KNASTER AND FRIENDS III: SUBADDITIVE COLORINGS

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 θ < k, the existence of a strongly unbounded coloring c: [k²] → θ is a theorem of 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: [k²] → θ is independent of 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 k-Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring c: [k²] → θ is equivalent to a certain weak indexed square principle ^ind(k, θ). We conclude the paper with an application to the failure of the infinite productivity of -stationarily layered posets, answering a question of Cox.