  Every better quasi-order codifies a Borel graph that does not contain a copy
of the shift graph. It is known that there is a better quasi-order that codes a
Borel graph with infinite Borel chromatic number, though one has yet to be
explicitly constructed. In this paper, we show that examples cannot be
constructed via standard methods. Moreover, we show that most of the known
better quasi-orders are non-examples, suggesting there is still a class of
better quasi-orders with interesting combinatorial properties who's
elements/members still remain unknown.
