Knaster and friends III: Subadditive colorings

0575310 - MÚ 2024 RIV GB eng J - Journal Article
Lambie-Hanson, Christopher - Rinot, A.
Knaster and friends III: Subadditive colorings.
Journal of Symbolic Logic. Roč. 88, č. 3 (2023), s. 1230-1280. ISSN 0022-4812. E-ISSN 1943-5886
Institutional support: RVO:67985840
Keywords : stationarily layered posets * strongly unbounded coloring * subadditive coloring
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Open access
https://doi.org/10.1017/jsl.2022.50

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 innite cardinals θ < κ, the existence of a strongly unbounded coloring c : [κ]2→ 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 : [κ]2→ 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 κ-Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring c : [κ]2→θ is equivalent to a certain weak indexed square principle ind(κ, 0). We conclude the paper with an application to the failure of the infinite productivity of κ-stationarily layered posets, answering a question of Cox.
Permanent Link: https://hdl.handle.net/11104/0345094