On compactness of weak square at singulars of uncountable cofinality

Abstract

Cummings, Foreman, and Magidor proved that Jensen's square principle is non-compact at $\aleph_ω$, meaning that it is consistent that $\square_{\aleph_n}$ holds for all $n<ω$ while $\square_{\aleph_ω}$ 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 $\square_κ^*$ 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 $\aleph_ω$.