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Cummings, Foreman, and Magidor proved that Jensen's square principle isnon-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 phenomenongeneralizes to singulars of uncountable cofinality. Surprisingly, we show thatunder some mild hypotheses, the weak square principle $\square_\kappa^*$ is infact compact at singulars of uncountable cofinality, and that an even strongerversion of these hypotheses is not enough for compactness of weak square at$\aleph_\omega$.

On compactness of weak square at singulars of uncountable cofinality