Abstract. In Reverse Mathematics, the axiom system DNR, asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0 (weak weak König's Lemma).

We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

Let r be a real number in the unit interval [0, 1]. A set A ⊆ ω is said to be coarsely computable at density r if there is a computable function f such that {n | f (n) = A(n)} has lower density at least r. Our main results are that A is coarsely computable at density 1/2 if A is either computably traceable or truth-table reducible to a 1-random set. In the other direction, we show that if a degree a is either hyperimmune or PA, then there is an a-computable set which is not coarsely computable at any positive density.

A set A ⊆ ω A\subseteq \omega is cototal if it is enumeration reducible to its complement,  A ¯ \overline {A} . The skip of  A A is the uniform upper bound of the complements of all sets enumeration reducible to  A A . These are closely connected:  A A has cototal degree if and only if it is enumeration reducible to its skip. We study cototality and related properties, using the skip operator as a tool in our investigation. We give many examples of classes of enumeration degrees that either guarantee or prohibit cototality. We also study the skip for its own sake, noting that it has many of the nice properties of the Turing jump, even though the skip of  A A is not always above  A A (i.e., not all degrees are cototal). In fact, there is a set that is its own double skip.

Abstract. We study the reverse mathematics and computability-theoretic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2-coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT

Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA 0 , to the failure of BΣ 0 2 .

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We construct the set of the title, answering a question of Cholak, Jockusch. and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of must contain a nonlow set.