3 problems

Benign cost-function conjecture. For each benign cost function $\mathbf{c}$, for each $B\models\mathbf{c}$, there is a c.e. set $A\models\mathbf{c}$ such that…

Benign cost-function conjecture. For each benign cost function $c$ and each set $B$ satisfying $B\models c$, there is a c.e. set $A$ satisfying $A\models c$ such that

Minimal-pair conjecture. There are cost functions $\mathbf{c}$ and $\mathbf{d}$ with the limit condition such that, for any $\Delta^0_2$ sets $A,B$,