UCLA

It is well known that the Borel chromatic number of a graph generated by a Borel function is omega or at most 3. In this dissertation we will prove that the Borel chromatic number of a graph generated by $n$ Borel functions that commute is omega or at most 2n+1. On top of that, we will prove that the Borel chromatic number for graphs generated by 2 functions is omega or at most 2*2 + 1 = 5, while the Borel chromatic number for graphs generated by 3 functions is omega or at most 8.