Universal countably chromatic graph

TL;DR

The paper proves the consistency of a universal object among countably chromatic graphs of size $\aleph_1$ together with $2^\omega=\aleph_2$. It achieves this by a finite-support forcing iteration of strongly proper ccc posets, starting from a model with $\,\mathsf{GCH}$, and by embedding a bookkeeping family of graphs into a growing universal structure via posets $\mathbb{P}_\delta$ and embeddings $f^p_\gamma$, controlled by ranks $\rho_\delta(\alpha)$, $\lambda_\delta(\alpha)$ and a labeling function $\ell$. The main technical contributions are the dense embedding arguments (density of the $f^G_\delta$ maps) and the strong properness/ccc analysis (trace residues and “super-nice” conditions) that preserve $\omega_1$ while forcing the continuum to $\aleph_2$. The results extend the universality/CH-interaction picture to countably chromatic graphs and generalize to any uncountable successor cardinal $\kappa^+$, illustrating a robust forcing approach to universality in non-elementary classes.

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 $κ^+$, where $κ$ is regular.

Theorems & Definitions (32)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Definition 3.1: $\mathbb{P}_1$
• Definition 3.4
• Definition 3.6: $\mathbb{P}_\delta$
• Lemma 3.7
• Lemma 4.1