Universal graphs between a strong limit singular and its power

TL;DR

This paper proves the consistency of a universal graph existing between a strong limit singular cardinal and its power, under certain large cardinal and forcing assumptions, impacting the understanding of universal structures in set theory.

Contribution

It establishes the consistency of a universal graph between a strong limit singular and its power, using forcing and large cardinal assumptions, addressing a longstanding problem.

Findings

Existence of a universal graph in the specified cardinality under certain conditions.

Failure of the Singular Cardinal Hypothesis at the singular cardinal.

Construction of a forcing extension with prescribed cofinality and universal graph.

Abstract

The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $\kappa$ is supercompact and the cardinals $\theta < \kappa$, $\lambda > \kappa$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(\kappa) = \theta$, the Singular Cardinal Hypothesis fails at $\kappa$ and there exists a universal graph in cardinality $\lambda \in (\kappa,2^\kappa)$.