In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility.

We call A weakly low for K if there is a c such that $K^A(\sigma )\ge K(\sigma )-c$ for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL₀. We study analogies and differences between WKL₀ and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

This paper addresses the strength of Ramsey's theorem for pairs (${\mathrm{R}\mathrm{T}}_2^2$) over a weak base theory from the perspective of 'proof mining'. Let ${\mathrm{R}\mathrm{T}}_2^{2-}$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of ${\Sigma }_1^0$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall \exists$-theorems.

There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles.

So from the perspective of proof unwinding the computational content of concrete proofs based on ${\mathrm{R}\mathrm{T}}_2^2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of ${\mathrm{R}\mathrm{T}}_2^2$ that are primitive recursive in f.

We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class ${\text{superhigh}}^{\diamond }$ and show that it contains some, but not all, of the noncomputable K-trivial sets.

For a subset A of a Polish group G, we study the (almost) packing index pack( A) (respectively, Pack( A)) of A, equal to the supremum of cardinalities |S| of subsets $S\subset G$ such that the family of shifts ${\left\{xA\right\}}_{x\in S}$ is (almost) disjoint (in the sense that for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are large in a geometric sense. We show that $\mathrm{pack}\left(A\right)\in \mathbb{N}\cup \left\{{\aleph }_0,\mathfrak{c}\right\}$ for any σ-compact subset A of a Polish group. In each nondiscrete Polish Abelian group G we construct two closed subsets $A,B\subset G$ with $\mathrm{pack}\left(A\right)=\mathrm{pack}\left(B\right)=\mathfrak{c}$ and $\text{Pack}(A\cup B)=1$ and then apply this result to show that G contains a nowhere dense Haar null subset $C\subset G$ with pack(C)=Pack(C)=κ for any given cardinal number $\kappa \in \left[4,\mathfrak{c}\right]$.

Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplicity, which are closely related to the study of automorphisms of the c.e. sets. We investigate the Turing degrees of these classes of c.e. sets. In particular, we show that the analogue of a theorem of Martin fails for these classes.