This paper will study sets and degrees containing sets that are determined as limits of computable approximations. By the Shoenfield Limit Lemma, the limit computable sets are precisely the degrees below ∅′. In particular, the paper will study limit computable sets by classifying them according to bounds to the number of changes to elements in various approximations of the sets. This leads to t...

Assuming ZFC, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω1 in the Turing degrees.

In this article we give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture on Borel functions from an analytic subset of a Polish space into a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to prove that several prominent results in recursion t...

by Stephen M. Walk We focus on a particular class of computably enumerable (c.e.) degrees, the array noncomputable degrees defined by Downey, Jockusch, and Stob, to answer questions related to array noncomputable degrees and definability in the partial ordering (R,≤) of c.e. degrees. First we demonstrate that the lattice M5 cannot be embedded into the c.e. degrees below every array noncomputabl...

Small independent edge dominating sets in graphs of maximum degree three – p. 1/34

Let d be a Turing degree, R[d] and Q[d] denote respectively classes of recursively enumerable (r.e.) and all degrees in which d is relatively enumerable. We proved in [ta] that there is a degree d containing differences of r.e.sets (briefly, d.r.e.degree) such that R[d] possess a least element m>0. Now we show the existence of a d.r.e. d such that R[d] has no a least element. We prove also that...

Alon, Kleitman, Lipton, Meshulam, Rabin and Spencer (Graphs. Combin. 7 (1991), no. 2, 97-99) proved, that for any hypergraph F = fF1; F2; : : : ; Fd(q 1)+1g, where q is a primepower, and d denotes the maximal degree of the hypergraph, there exists an F0 F , such that jSF2F0 F j 0 (mod q). We give a direct, alternative proof for this theorem, and we also give an explicit construction of a hyperg...

This paper is concerned with the construction of intervals of computably enumerable degrees in which the lattice M5 (see Figure 1) cannot be embedded. Actually, we construct intervals I of computably enumerable degrees without any weak critical triples (this implies that M5 cannot be embedded in I, see Section 2). Our strongest result is that there is a low2 computably enumerable degree e such ...

It is investigated how much information of a random set can be preserved if one splits the random set into two halves or, more generally, cuts out an infinite portion with an infinite recursive set. The two main results are the following ones: 1. Every high Turing degree contains a Schnorr random set Z such that Z ≡T Z ∩ R for every infinite recursive set R. 2. For each set X there is a Martin-...

It is shown that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, this answers negatively a recent question on the characterization of these sets. Furthermore, it is shown that every set which is low for weakly 1-generic is also low for Kurtz-random. In addition to this, it is shown that a se...