An antibasis result for graphs of infinite Borel chromatic number

We answer in the negative a question posed by Kechris-Solecki-Todorcevic as to whether the shift graph on Baire space is minimal among graphs of indecomposably infinite Borel chromatic number. To do so, we use ergodic-theoretic techniques to construct a new graph amalgamating various properties of the shift actions of free groups. The resulting graph is incomparable with any graph induced by a function. We then generalize this construction and collect some of its useful properties.