A Turing Incomputable Coloring Function

This paper describes a sequence of natural numbers that grows faster than any Turing computable function. This sequence is generated from a version of the tiling problem, called a coloring system. In our proof that generates the sequence, we use the notions of a chain and an unbounded sequence property, which resemble the methods of point set topology. From this sequence, we define a Turing incomputable coloring function.