Weak Borel chromatic numbers

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G-independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge. We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most א1. We observe that some weak version of Todorcevic’s Open Coloring Axiom for closed colorings follows from MA. Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most א1. We refute various reasonable generalizations of these results to hypergraphs.