On compactness of weak square at singulars of uncountable cofinality

TL;DR

This paper explores the compactness properties of the weak square principle at singular cardinals with uncountable cofinality, revealing conditions under which it is or isn't compact.

Contribution

It demonstrates that under certain hypotheses, weak square is compact at uncountable cofinality singulars, contrasting with known results at $eth_\omega$.

Findings

Weak square is compact at singulars of uncountable cofinality under mild hypotheses.

Stronger hypotheses do not guarantee compactness at $eth_\omega$.

The phenomenon at $eth_\omega$ does not generalize straightforwardly to all uncountable cofinality singulars.

Abstract

Cummings, Foreman, and Magidor proved that Jensen's square principle is non-compact at $\aleph_\omega$, meaning that it is consistent that $\square_{\aleph_n}$ holds for all $n<\omega$ while $\square_{\aleph_\omega}$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle $\square_\kappa^*$ is in fact compact at singulars of uncountable cofinality, and that an even stronger version of these hypotheses is not enough for compactness of weak square at $\aleph_\omega$.