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

Abstract

We prove that consistently, cov($\mathcal{M})<{\lambda }_0<{\lambda }_1<{\lambda }_{\infty }<2^{{\aleph }_0}$, where ${\lambda }_0$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todorčević graph ${\mathbb{G}}_0$, that is, the minimal cardinality of a ${\mathbb{G}}_0$-independent Borel covering of $2^{\omega }$, while ${\lambda }_1$ and ${\lambda }_{\infty }$ are the corresponding invariants of the graph ${\mathbb{G}}_1$ and the simple graph associated with the equivalence relation $E_0$.