  In this work we study the uncountable Borel chromatic numbers, defined by
Geschke (2011) as cardinal characteristics of the continuum, of low complexity
graphs. We show that a strong form of locally countable graphs with compact
totally disconnected set of vertices have Borel chromatic number bounded by the
continuum of the ground model in the model obtained by adding $\aleph_2$ Laver
reals. From this, we answer a question from Geschke and the second author
(2022), and another question from Fisher, Friedman and Khomskii (2014)
concerning regularity properties of subsets of the real line.
