Universal graphs between a strong limit singular and its power

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 $\text{cf}(\kappa )=\theta$, the Singular Cardinal Hypothesis fails at $\kappa$ and there exists a universal graph in cardinality $\lambda \in (\kappa ,2^{\kappa })$.