Knaster and friends

יום ב', 19/03/2018 - 13:00

In the 1970's, consistent examples of k-cc posets whose square is not k-cc were constructed by Laver, Fleissner, and Galvin. Later on, ZFC examples were constructed by Todorcevic, Shelah and others. The hardest case, being k=w2, was resolved by Shelah in 1997.
In this work, we obtain analogous results for k-Knaster posets. Among others, for any successor cardinal k, we produce a ZFC example of a k-Knaster poset whose w-power is not k-cc.
To do so, we introduce a new coloring principle, and establish the existence of various instances of it.
We also introduce a new cardinal invariant for k, denoted chi(k), that, roughly speaking, measures how far k is from being weakly compact. It is proved that by forcing over a model with a weakly compact cardinal k, chi(k) could be made equal to any prescribed regular cardinal <= k.
Further byproducts of this work show that the main results of [1] and [2] are sharp.

This is joint work with Chris Lambie-Hanson.

[1] A. Rinot, Transforming rectangles into squares, with applications to strong colorings, Adv. Math., 231(2): 1085-1099, 2012.
[2] A. Rinot, Complicated colorings, Math. Res. Lett., 21(6): 1367–1388, 2014.