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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?

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  • $\begingroup$ Interesting question +1. Your title refers to "meeting and avoid computable sets", but the property is about computable sets meeting and avoiding $X$. $\endgroup$
    – Joel David Hamkins
    Commented Jan 5, 2023 at 2:13
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    $\begingroup$ Do you know what happens with a Sacks-generic real? $\endgroup$
    – Joel David Hamkins
    Commented Jan 5, 2023 at 2:44
  • $\begingroup$ A Sacks-generic real won't itself be hesive, but it might compute one. I'll think on that a bit. $\endgroup$
    – Dan Turetsky
    Commented Jan 5, 2023 at 2:55
  • $\begingroup$ I was thinking that perhaps a fusion argument would enable you to prove it also couldn't compute one. Only countably many programs, and the fusion argument can handle them one at a time. $\endgroup$
    – Joel David Hamkins
    Commented Jan 5, 2023 at 2:57
  • $\begingroup$ "A set"... you mean, a subset of the set of integers? $\endgroup$
    – YCor
    Commented Jan 5, 2023 at 21:00

1 Answer 1

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$X$ 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, Z. Math. Logik Grundlagen Math. 15, 135-140 (1969). ZBL0184.02002.

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

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    $\begingroup$ Does he use a fusion argument? $\endgroup$
    – Joel David Hamkins
    Commented Jan 5, 2023 at 19:12
  • $\begingroup$ @JoelDavidHamkins I don't think he advertised it as such, but maybe. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Jan 5, 2023 at 19:15
  • $\begingroup$ With a Sack real, I would think he did, since that is the main thing to do with Sacks reals and the trees approximating them. $\endgroup$
    – Joel David Hamkins
    Commented Jan 5, 2023 at 19:17
  • $\begingroup$ @JoelDavidHamkins I think Spector constructed a minimal Turing degree using "uniform" trees and those don't suffice to force bi-immunity. Can you do fusion arguments with those trees? $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Jan 5, 2023 at 19:23
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    $\begingroup$ @JoelDavidHamkins Thanks, Richard Shore told me about that at some point but a reminder was helpful. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Jan 5, 2023 at 19:33

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