Exact Pairs for the Ideal of the k-Trivial Sequences in the Turing Degrees

The K-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later in [Nie09, Problem 5.5.8]. We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a K-trivial degree d such that for all degrees a,b which are not K-trivial and a > d,b > d there exists a degree v which is not K-trivial and a > v,b > v. This work sheds light to the question of the definability of the K-trivial degrees in the c.e. degrees.