A complexity problem for Borel graphs

We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on $[\mathbb{N}]^{<\mathbb{N}}$ with finite (or, equivalently, $\leq 3$) Borel chromatic number form a $\mathbf{\Sigma}^1_2$-complete set. This answers a question of Kechris and Marks and strengthens several earlier results.