Open Access Publications from the University of California

## The Interaction Between Weak Variants of Square and Other Combinatorial Principles in Set Theory

• Author(s): Susice, John Peter
We investigate weak variants of Jensen's square principle $\square_{\kappa}$ and show that there are a variety of set-theoretic principles which although inconsistent with $\square_{\kappa}$, are nonetheless consistent with one of its weak variants.
It is well-known that $\square_{\omega_1}$ is inconsistent with Chang's Conjecture. Sakai, however, showed that \square_{\omega_1, 2}$is compatible with Chang's Conjecture, assuming the existence of a measurable cardinal \cite{cc_weak_square}. In light of this, he posed the question of the exact consistency strength of this conjunction. We answer this question by pushing down Sakai's large cardinal hypothesis to an$\omega_1$-Erd\H{o}s cardinal, which is optimal due to work of Silver and Donder. Shelah and Stanley showed that for$\kappa$uncountable,$\square_{\kappa}$implies the existence of a non-special$\kappa^+$-Aronszajn tree \cite{shelah_stanley}. We show that this result is best possible in the sense that for any regular$\kappa$,$\square_{\kappa, 2}$is consistent with all$\kappa^+$-Aronszajn trees are special'' (assuming the existence of a weakly compact cardinal). Moreover, by employing methods of Golshani and Hayut, we are able to establish this consistency result simultaneously for all regular$\kappa$from the existence of class many supercompact cardinals. Finally, we introduce a weak variant$R_2^*(\aleph_2, \aleph_1)$of the reflection principle$R_2(\aleph_2, \aleph_1)$introduced by Rinot and show that unlike Rinot's principle our weak variant is consistent with$\square(\omega_2)$(though still inconsistent with$\square_{\omega_1}\$).