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 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.
Original language | English |
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Pages (from-to) | 1230-1280 |
Number of pages | 51 |
Journal | Journal of Symbolic Logic |
Volume | 88 |
Issue number | 3 |
DOIs | |
State | Published - 30 Sep 2023 |
Externally published | Yes |
Keywords
- Aronszajn tree
- ascent path
- coherent coloring
- indexed square
- stationarily layered posets
- strongly unbounded coloring
- subadditive coloring
ASJC Scopus subject areas
- Philosophy
- Logic