A general tool for consistency results related to I1

Abstract

In this paper we provide a general tool to prove the consistency of $I1(\lambda )$ with various combinatorial properties at $\lambda$ typical at settings with $2^{\lambda }>{\lambda }^{+}$, that does not need a profound knowledge of the forcing notions involved. Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at ${\lambda }^{+}$ and ${\lambda }^{++}$.