Singular Cardinals and Square Properties Menachem Magidor and Dima Sinapova

We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, for some regular cardinal ν. More precisely, we show that if V ⊂ W , κ < ν are cardinals, where ν is regular in V , κ is a singular cardinal in W of countable cofinality, cf (τ) = ω for all V -regular κ ≤ τ ≤ ν, and (ν) = (κ) , then W |= κ,ω.