# 2010 seminar talk: What Andrew and I talked about last week

Talk held by Sy-David Friedman (KGRC) at the KGRC seminar on 2010-05-27.

### Abstract

Actually I will only tell you about the mathematics we discussed, saving our discussion
of kangaroos for a later time. We looked at the interplay between Square/Stationary Reflection
on the one hand, and Large Cardinal Axioms on the other. Solovay showed ages ago that supercompactness
kills Square, improved recently by Jensen, who showed that if κ is Subcompact then Square_{κ} fails.
We extend Jensen's result to: If κ is α^{+}-Subcompact (and κ is at most α) then Square_{α}
(indeed Square_{α,<κ} fails. Moreover, forcing shows that this is best possible: One can preserve
all instances of α-Subcompactness as well as very LARGE cardinals (like ω-Superstrongs) and force
Square_{α} to hold everywhere not ruled out by the previous result. We found similar results with Square_{α} replaced
by Stationary Reflection at α^{+} (on small cofinalities) and with α^{+} Subcompactness replaced by α^{++}
Subcompactness. The proof of this latter result, unlike the proofs of the earlier ones, caused us some worries.