Abstract
The Borel chromatic number - introduced by Kechris, Solecki, and
Todorcevic (1999) - generalizes the chromatic number on finite graphs
to definable graphs on topological spaces. While the G0 dichotomy
states that there exists a minimal graph with uncountable Borel
chromatic number, it turns out that characterizing when a graph has
infinite Borel chromatic number is far more intricate. Even in the
case of graphs generated by a single function, our understanding is
actually very poor. The Shift Graph on the space of infinite subsets
of natural numbers is generated by the function that removes the
minimum element. It is acyclic but has infinite Borel chromatic
number. In 1999, Kechris, Solecki, and Todorcevic asked whether the
Shift Graph is minimal among the graphs generated by a single Borel
function that have infinite Borel chromatic number. I will explain why
the answer is negative using a representation theorem for
Σ12 sets due to Marcone.