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


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 $\square_{\omega_1}$).

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