Given a graph G = (V, E), we define the transpose degree sequence d T j to be equal to the number of vertices of degree at least j. We define L G , the graph Laplacian, to be the matrix, whose rows and columns are indexed by the vertex set V , whose diagonal entry at v is the degree of v and whose value at a pair (v, w) is −1 if (v, w) ∈ E and 0 otherwise. Grone and Merris conjectured [GM] Conj...

We study here the degree-theoretic structure of set-theoretical splittings of recursively enumerable (r.e.) sets into differences of r.e. sets. As a corollary we deduce that the ordering of wtt–degrees of unsolvability of differences of r.e. sets is not a distributive semilattice and is not elementarily equivalent to the ordering of r.e. wtt–degrees of unsolvability.

Sacks [14] showed that every computably enumerable (c.e.) degree ≥ 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. sp...

We show the undecidability of the Π3-theory of the partial order of computably enumerable Turing degrees.

Slaman and Woodin have developed and used set-theoretic methods to prove some remarkable theorems about automorphisms of, and de…nability in, the Turing degrees. Their methods apply to other coarser degree structures as well and, as they point out, give even stronger results for some of them. In particular, their methods can be used to show that the hyperarithmetic degrees are rigid and biinter...

The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become the longest-standing open question concerning the structure of the computably enumerable (c.e.) degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, 0′ = b ∨ x if and only if 0′ = a ∨ x. In this paper, we show that every c.e. degree b 6= 0 or 0′ has a major subdegree, answerin...

The existence of minimal degrees is investigated for several polynomial reducibilities. It is shown that no set has minimal degree with respect to polynomial many-one or Turing reducibility. This extends a result. of Ladner [L] whew reciirsive sets are considered. An "honest '' polynomial reducibility, < ; , is defined which is a strengthening of polynomial Turing reduc-ibility. We prove that n...

Let k be a finite extension of Qp and Ek be the set of the extensions of degree p over k whose normal closure is a p-extension. For a fixed discriminant, we show how many extensions there are in EQp with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalize...

A phylogenetic tree on a label set L is a tree which exactly ILl leaves (vert.ices of degree one), no vertices of degree two, and each leaf labeled with a distinct element from L. A binary phylogenetic tree is a phylogenetic tree in which every non-leaf has degree three. Phylogenetic trees get their name because they are often appropriate models for evolutionary history. However, for simplicity...

Claim 1.1. Let L be a set of n lines in R. Then there exists a nontrivial polynomial f ∈ R[x1, x2, x3] of degree smaller than 3 √ n that vanishes on all the lines of L. Proof. Let P be a set of at most 4n points, that is obtained by arbitrarily choosing 4 √ n points from every line of L. Since ( 3 √ n+3 3 ) > 4n, by Lemma 2.1 of Chapter 5 there exists a nontrivial polynomial f ∈ R[x1, x2, x3] o...