The ineffable tree property and failure of the singular cardinals hypothesis

Abstract

ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal κ \kappa , ITP at κ \kappa holds if and only if κ \kappa is supercompact. And just like the tree property, it can be forced to hold at accessible cardinals. A broad project is obtaining ITP at many cardinals simultaneously. Past a singular cardinal, this requires failure of SCH. We prove that from large cardinals, it is consistent to have failure of SCH at κ \kappa together with ITP κ + \kappa ^+ . Then we bring down the result to κ = ℵ ω 2 \kappa =\aleph _{\omega ^2} .