Cone Avoidance and the No-Least-Join Theorem

Abstract

We give a column-avoidance modification of the no-least-join construction of Downey, Greenberg, Lewis, and Montalbán. We show that if $C$ is computably enumerable, $A,B{<}_{T}C$, and $A/{\le }_{T}B$, then for every noncomputable set $E$ there is a set $X{\le }_{T}C$ such that $B{\le }_{T}A\oplus X$ and $E/{\le }_{T}X$. Consequently, for the corresponding Turing degrees $a,b,c$, the degrees $x\le c$ satisfying $b\le a\vee x$ have no nonzero common lower bound; equivalently, no nonzero Turing cone contains all of them. This gives a negative answer to their Question 1.19.

arXiv comment15 pages