Small u(kappa) at singular kappa with compactness at kappa++

TL;DR

This paper demonstrates the consistency of certain combinatorial properties at the double successor of a singular strong limit cardinal with the continuum function, using advanced forcing techniques.

Contribution

It establishes the consistency of the tree property, stationary reflection, and the failure of approachability at b2 with a small b6, expanding understanding of large cardinal combinatorics.

Findings

Tree property, stationary reflection, and failure of approachability are consistent at b2 with b6 = b1 < 2^.

Stationary reflection at b1 is indestructible under all -cc forcings.

Results apply to both countable and uncountable cofinalities of .

Abstract

We show that the tree property, stationary reflection and the failure of approachability at $\kappa^{++}$ are consistent with $\mathfrak{u}(\kappa) = \kappa^+ < 2^\kappa$, where $\kappa$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $\lambda$ is a regular cardinal, then stationary reflection at $\lambda^+$ is indestructible under all $\lambda$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).