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Soare proved that the maximal sets form an orbit in $mathcal{E}$. We consider here $mathcal{D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer. Some orbits of $mathcal{D}$-maximal sets are well understood, e.g., hemimaximal sets, but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the $mathcal{D}$-maximal sets. Although these invariants help us to better understand the $mathcal{D}$-maximal sets, we use them to show that several classes of $mathcal{D}$-maximal sets break into infinitely many orbits.
We provide a proof that analytic almost disjoint families of infinite sets of integers cannot be maximal using a result of Bourgain about compact sets of Baire class one functions. Inspired by this and related ideas, we then provide a new proof of th
In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a
It is shown, from hypotheses in the region of $omega^2$ Woodin cardinals, that there is a transitive model of KP + AD$_mathbb{R}$ containing all reals.
There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(Delta + log^* n)$ communication rounds; here $n$ is the number of nodes and $Delta$ is the maximum degree. The lower bound by Linial (1987, 1992)
Harrington and Soare introduced the notion of an n-tardy set. They showed that there is a nonempty $mathcal{E}$ property Q(A) such that if Q(A) then A is 2-tardy. Since they also showed no 2-tardy set is complete, Harrington and Soare showed that the