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

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.Comment: 19 page

The degree structure of Weihrauch-reducibility

We answer a question by Vasco Brattka and Guido Gherardi by proving that the Weihrauch-lattice is not a Brouwer algebra. The computable Weihrauch-lattice is also not a Heyting algebra, but the continuous Weihrauch-lattice is. We further investigate the existence of infinite infima and suprema, as well as embeddings of the Medvedev-degrees into the Weihrauch-degrees

Inactivation of tumor suppressor Dlg1 augments transformation of a T-cell line induced by human T-cell leukemia virus type 1 Tax protein

BACKGROUND: The interaction of human T-cell leukemia virus type 1 (HTLV-1) Tax1 protein with the tumor suppressor Dlg1 is correlated with cellular transformation. RESULTS: Here, we show that Dlg1 knockdown by RNA interference increases the ability of Tax1 to transform a mouse T-cell line (CTLL-2), as measured interleukin (IL)-2-independent growth. A Tax1 mutant defective for the Dlg1 interaction showed reduced transformation of CTLL-2 compared to wild type Tax1, but the transformation was minimally affected by Dlg1 reduction. The few Tax1ΔC-transduced CTLL-2 cells that became transformed expressed less Dlg1 than parental cells, suggesting that Dlg1-low cells were selectively transformed by Tax1ΔC. Moreover, all human T-cell lines immortalized by HTLV-1, including the recombinant HTLV-1-containing Tax1ΔC, expressed less Dlg1 than control T-cell lines. CONCLUSION: These results suggest that inactivation of Dlg1 augments Tax1-mediated transformation of CTLL-2, and PDZ protein(s) other than Dlg1 are critically involved in the transformation

Propagation of partial randomness

Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree