On density of old sets in Prikry type extensions

Abstract:

Every set of ordinals of cardinality $\kappa$ in a Prikry extension with a measure over $\kappa$ contains an old set of arbitrarily large cardinality below $\kappa$, and, actually, it can be split into countably many old sets. What about sets with larger cardinalities? Clearly, any set of ordinals in a forcing extension of a regular cardinality above the cardinality of the forcing used, contains an old set of the same cardinality. Still cardinals in the interval $(\kappa , 2^\kappa ]$ remain. Here we would like to address this type of question.