On the weak Borel chromatic number and cardinal invariants of the continuum

TL;DR

This paper explores the relationships between the weak Borel chromatic number and certain cardinal invariants of the continuum, establishing consistent inequalities among them using advanced set-theoretic methods.

Contribution

It demonstrates the consistent strict inequalities among the weak Borel chromatic number and specific continuum cardinal invariants for particular graphs.

Findings

Established that cov($\\mathcal{M})$ can be less than the weak Borel chromatic numbers of certain graphs.

Proved the existence of models where the invariants satisfy strict inequalities.

Connected graph invariants with classical cardinal characteristics of the continuum.

Abstract

We prove that consistently, cov($\mathcal{M})< \lambda_\mathbf{0} < \lambda_\mathbf{1} < \lambda_\mathbf{\infty} < 2^{\aleph_0}$, where $\lambda_\mathbf{0}$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todor\v{c}evi\'c graph $\mathbb{G}_0$, that is, the minimal cardinality of a $\mathbb{G}_0$-independent Borel covering of $2^\omega$, while $\lambda_\mathbf{1}$ and $\lambda_\infty$ are the corresponding invariants of the graph $\mathbb{G}_1$ and the simple graph associated with the equivalence relation $E_0$.