Martin—Löf (ML)-reducibility compares the complexity of K-trivial sets of natural numbers by examining the Martin—Löf random sequences that compute them. One says that a K-trivial set A is ML-reducible to a K-trivial set B if every ML-random computing B also computes A. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets intended to capture ML-random sequences. We show that for every cost function there is a c.e. set that is ML-complete among the sets obeying it. We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω given by a computable set of bit positions. This leads to a new characterisation of strong jump-traceability.
In this series of papers we advance Ramsey theory over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin’s Axiom: a function p: [ω1]2 → ω witnesses a weak negative Ramsey relation when p plays the role of a coloring if and only if a positive Ramsey relation holds over p when p plays the role of a partition.
The consistency of positive Ramsey relations over partitions does not stop at the first uncountable cardinal: it is established that at arbitrarily high uncountable cardinals these relations follow from forcing axioms without large cardinal strength. This result solves in particular two problems from [CKS21].
In this paper we study speedups of dynamical systems in the topological category. Specifically, we characterize when one minimal homeomorphism on a Cantor space is the speedup of another. We go on to provide a characterization for strong speedups, i.e., when the jump function has at most one point of discontinuity. These results provide topological versions of the measure-theoretic results of Arnoux, Ornstein and Weiss, and are closely related to Giordano, Putnam and Skau’s characterization of orbit equivalence for minimal Cantor systems.
In this series of papers we advance Ramsey theory over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21].
It is shown that for every infinite cardinal λ, a strong coloring on λ+ by λ colors over a partition can be stretched to one with λ+ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing Pr1(…) over a partition may be improved to witness Pr0(…).
Since the classical theory corresponds to the special case of a partition with just one cell, the two results generalize pump-up theorems due to Eisworth and Shelah, respectively.
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