ON COMPACTNESS OF WEAK SQUARE AT SINGULARS OF UNCOUNTABLE COFINALITY

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 ${{\mathsf {PCF}}}$-theoretic hypotheses, the weak square principle $\square _\kappa ^*$ is in fact compact at singulars of uncountable cofinality.