  A long-standing conjecture of Sacks states that it is provable in ZFC that
every locally countable partial order of size continuum embeds into the Turing
degrees. We show that this holds for partial orders of height two, but provide
evidence that it is hard to extend this result even to partial orders of height
three. In particular, we show that the result for height two partial orders
holds both in certain extensions of ZF with only limited forms of choice and in
the Borel setting (where the partial orders and embeddings are required to be
Borel measurable), but that the analogous result for height three partial
orders fails in both of these settings. We also formulate a general obstacle to
embedding partial orders into the Turing degrees, which explains why our
particular proof for height two partial orders cannot be extended to height
three partial orders, even in ZFC. We finish by discussing how our results
connect to the theory of countable Borel equivalence relations.
