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An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $mathcal H^s$ contains a closed subset of non-zero (and indeed finite) $mathcal H^s$-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) $Sigma^1_1$ set of reals in Cantor space, there is always a $Pi^0_1(mathcal{O})$ subset on non-zero $mathcal H^s$-measure definable from Kleenes $mathcal O$. On the other hand, there are $Pi^0_2$ sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.
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The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. For classical atomic sets, we study whether they contain or not infinite uniformly supported subsets.
We study reductions well suited to compare structures and classes of structures with respect to properties based on enumeration reducibility. We introduce the notion of a positive enumerable functor and study the relationship with established reductions based on functors and alternative definitions.
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 maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size $aleph_{alpha}$, then the set has size $aleph_{alpha}$ if $aleph_{alpha}$ is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If $G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is an ordering $<$ of $V_{G}$ such that ${w < v : {w,v} in E_{G}}$ is a clique for each $vin V_{G}$.
We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2. If in a partially ordered set, all chains are finite and all antichains have size $aleph_{alpha}$, then the set has size $aleph_{alpha}$ for any regular $aleph_{alpha}$. 3. CS (Every partially ordered set without a maximal element has two disjoint cofinal subsets). 4. CWF (Every partially ordered set has a cofinal well-founded subset). 5. DT (Dilworths decomposition theorem for infinite p.o.sets of finite width). 6. If the chromatic number of a graph $G_{1}$ is finite (say $k<omega$), and the chromatic number of another graph $G_{2}$ is infinite, then the chromatic number of $G_{1}times G_{2}$ is $k$. 7. For an infinite graph $G=(V_{G}, E_{G})$ and a finite graph $H=(V_{H}, E_{H})$, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$. Further we study a few statements restricted to linearly-ordered structures without the axiom of choice.
In this talk we discuss the microscopic limit of QCD at nonzero chemical potential. In this domain, where the QCD partition function is under complete analytical control, we uncover an entirely new link between the spectral density of the Dirac operator and the chiral condensate: violent complex oscillations on the microscopic scale give rise to the discontinuity of the chiral condensate at zero quark mass. We first establish this relation exactly within a random matrix framework and then analyze the importance of the individual modes by Fourier analysis.
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