Small universal families of graphs on $\aleph_{\omega+1}$

TL;DR

This paper demonstrates the consistency of having a strong limit cardinal _, a large power set, and a small universal family of graphs on _{+1} using advanced forcing techniques.

Contribution

It introduces a method to simultaneously control the size of the continuum and the universality number of graphs on _{+1} via Prikry forcing with interleaved collapsing.

Findings

_ is a strong limit in the model.

The continuum ^{_} can be made arbitrarily large.

The universality number for graphs on _{+1} is small in the constructed model.

Abstract

We prove that it is consistent that $\aleph_\omega$ is strong limit, $2^{\aleph_\omega}$ is large and the universality number for graphs on $\aleph_{\omega+1}$ is small. The proof uses Prikry forcing with interleaved collapsing.