A Note on a Conjecture of Sacks: It is Harder to Embed Height Three Partial Orders than Height Two Partial Orders

TL;DR

This paper investigates the complexity of embedding partial orders into Turing degrees, showing that height two cases are manageable under various set-theoretic assumptions, but height three cases present significant obstacles.

Contribution

It proves that height two partial orders can be embedded into Turing degrees in ZF extensions and Borel settings, but demonstrates the failure of this for height three, revealing a fundamental complexity gap.

Findings

Height two partial orders embed into Turing degrees in ZF extensions.

Embedding height three partial orders into Turing degrees fails in Borel and limited choice settings.

A general obstacle explains why height three embeddings are fundamentally harder.

Abstract

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.