The extension of embeddings problem for the recursively enumer-able degrees R = (R; <; 0;0 0) asks for given nite partially ordered sets P Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an algebraic insight into R assert either extension or nonextension of embeddings. We extend, strengthen...

A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A. That is, R is not named in the language of A. We define DgA(R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models. We investigate conditions on A and R...

It is shown that for any 2-computably enumerable Turing degree l, any computably enumerable degree a, and any Turing degree s, if l′ = 0′, l < a, s ≥ 0′, and s is c.e. in a, then there is a 2-computably enumerable degree x with the following properties: (1) l < x < a, and (2) x′ = s.

The goal of this paper is to define fermionic fields on causal set. This is done by the use of holonomies to define vierbines, and then defining spinor fields by taking advantage of the leftover degrees of freedom of holonomies plus additional scalar fields. Grassmann nature is being enforced by allowing measure to take both positive and negative values, and also by introducing a vector space t...

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing. More precisely, let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-packing. We show that

A Turing degree a is said to be almost everywhere dominating if, for almost all X ∈ 2 with respect to the “fair coin” probability measure on 2, and for all g : ω → ω Turing reducible to X, there exists f : ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We r...

For k > 2, let H be a k-uniform hypergraph on n vertices and m edges. Let S be a set of vertices in a hypergraph H. The set S is a transversal if S intersects every edge of H, while the set S is strongly independent if no two vertices in S belong to a common edge. The transversal number, τ(H), of H is the minimum cardinality of a transversal in H, and the strong independence number of H, α(H), ...

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem [12] characterises the minimum degree that ensures a graph G contains a perfect Kr-packing. Balogh, Kostochka and Treglown [4] proposed a degree sequence version of the Hajnal–Szemerédi theorem which, if true, gives a strength...