Knaster and friends III: Subadditive colorings

Joint work with Chris Lambie-Hanson.

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 $c:[\kappa {]}^2\to \theta$ 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 $c:[\kappa {]}^2\to \theta$ 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 $c:[\kappa {]}^2\to \theta$ is equivalent to a certain weak indexed square principle ${\boxminus }^{\mathrm{ind}}(\kappa ,\theta )$.

We conclude the paper with an application to the failure of the infinite productivity of $\kappa$-stationarily layered posets, answering a question of Cox.

Downloads: