Universal graphs at $\aleph_{\omega_1+1}$

TL;DR

This paper constructs a model of set theory starting from a supercompact cardinal where the continuum sizes are controlled, and a small jointly universal family of graphs exists on a large cardinal, demonstrating a new method for such constructions.

Contribution

It introduces a novel technique to build models with specified continuum sizes and small universal families of graphs at uncountable cardinals, generalizing previous results.

Findings

Existence of a model with controlled continuum sizes starting from a supercompact cardinal.

Construction of a small jointly universal family of graphs on a large uncountable cardinal.

Technique applicable to any uncountable cardinal, not just .

Abstract

Starting from a supercompact cardinal we build a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$. The same technique will work for any uncountable cardinal in place of $\omega_1$.