Call a set $X$ *hesive* if for every infinite computable set $C$, both $C \cap X$ and $C \setminus X$ are infinite.


It's not hard to see that every hyperimmune degree computes a hesive set, but this isn't a characterization, since also any random set is hesive (in fact, Church stochasticity suffices).


Does every noncomputable degree compute a hesive set?



Sets meeting and avoiding computable sets
$X$ is hesive iff $X$ is bi-immune.


Jockusch showed that a Sacks generic has bi-immune-free degree.


*Jockusch, C. G. Jr.*, [**The degrees of bi-immune sets**](http://dx.doi.org/10.1002/malq.19690150707), Z. Math. Logik Grundlagen Math. 15, 135-140 (1969). [ZBL0184.02002](https://zbmath.org/?q=an:0184.02002).


So no, not every noncomputable degree contains a hesive set.


