On the weak Borel chromatic number and cardinal invariants 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 $E_0$.