The concept of an $L$-bornology is introduced and the theory of $L$-bornological spacesis being developed. In particular the lattice of all $L$-bornologies on a given set is studied and basic properties ofthe category of $L$-bornological spaces and bounded mappings are investigated.

Introduction. According to a classical theorem of Post, the arithmetical sets may be obtained by the following construction : Step 0: "take" all the r.e. sets. Step n+1 : "add" all sets which are r.e. in sets "taken" at a previous stage. Moreover, this construction is intimately related to the Kleene arithmetical hierarchy, defined in terms of the number and quality of alternating numberquantif...

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...

We study sufficient l-degree (1 ≤ l < k) conditions for the appearance of perfect and nearly perfect matchings in k-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition (l = 1) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on n vertices with minimum vertex degree at least (5/9+o(1)) ` n 2 ́ contains a perfect matchi...

We consider a variant of the game of Cops and Robbers, called Containment, in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). The cops win by “containing” the robber, that is, by occupying all edges incident with a vertex occupied by the robber. The minimum number of cops, ξ(G), required to contain a ...

Π1 classes are important to the logical analysis of many parts of mathematics. The Π1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relatin...

A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show t...

In Chapter I, we exhibit a high strongly noncappable degree. Chapter II answers negatively the question whether a deep degree exists. It also shows a weak converse of this. Chapter III is devoted to index sets. We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. We also classify the index set of quasimax...