Sacks's Conjecture on embedding partial orders in the Turing degrees

A partial order $(P, \leq)$ is locally countable if for all $p \in P$, the set $\{q \in P \mid q \leq p\}$ is countable.

• Sacks proved every locally countable partial order of size $\aleph_1$ embeds into the Turing degrees. Hence the conjecture is true under $\mathsf{CH}$.
• Higuchi showed that every locally countable partial order of size continuum and height two embeds into the Turing degrees.
• Miller and Greenberg showed that every locally countable partial order of size continuum and height three which has at most $\aleph_1$ elements of depth three embeds into the Turing degrees.

The conjecture was first stated by Gerald Sacks in his 1963 book Degrees of Unsolvability. It was listed there as one of 6 conjectures, the rest of which have since been solved.