Knaster and friends III: Subadditive colorings

TL;DR

This paper investigates the existence and independence of subadditive strongly unbounded colorings in set theory, linking them to various infinitary combinatorial principles and providing applications to poset productivity.

Contribution

It establishes the independence of subadditive strongly unbounded colorings from ZFC and connects them to key combinatorial principles, extending previous results on unbounded colorings.

Findings

Existence of subadditive strongly unbounded colorings is often independent of ZFC.

Such colorings are equivalent to certain weak indexed square principles.

Application to the failure of infinite productivity of certain posets.

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 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 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 $\kappa$-Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring is equivalent to a certain weak indexed square principle. We conclude the paper with an application to the failure of the infinite productivity of $\kappa$-stationarily layered posets, answering a question of Cox.