Universal countably chromatic graph

TL;DR

This paper demonstrates the consistency of a universal countably chromatic graph of size 91 with the failure of the continuum hypothesis, using a forcing iteration of strongly proper ccc posets.

Contribution

It introduces a method to construct a universal countably chromatic graph of size 91 under certain set-theoretic assumptions, extending previous results.

Findings

Existence of such a universal graph is consistent with the failure of CH.

The construction uses a forcing iteration of strongly proper ccc posets.

Applicable to any uncountable successor cardinal 97, where 9 is regular.

Abstract

We show that the existence of a universal countably chromatic graph of size $\aleph_1$ together with the failure of continuum hypothesis is consistent. The proof is a forcing iteration of strongly proper ccc posets. The construction works for any uncountable successor cardinal $\kappa^+$, where $\kappa$ is regular.