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Benjamin Miller
University of Illinois, Urbana-Champaign
  1.  43
    The Graph-Theoretic Approach to Descriptive Set Theory.Benjamin D. Miller - 2012 - Bulletin of Symbolic Logic 18 (4):554-575.
    We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.
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  2.  12
    An Embedding Theorem Of.Itay Kaplan & Benjamin D. Miller - 2014 - Journal of Mathematical Logic 14 (2):1450010.
    We provide a new criterion for embedding.
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  3.  22
    Basis Theorems for Non-Potentially Closed Sets and Graphs of Uncountable Borel Chromatic Number.Dominique Lecomte & Benjamin D. Miller - 2008 - Journal of Mathematical Logic 8 (2):121-162.
    We show that there is an antichain basis for neither the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.
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  4.  34
    Dichotomy Theorems for Countably Infinite Dimensional Analytic Hypergraphs.Benjamin D. Miller - 2011 - Annals of Pure and Applied Logic 162 (7):561-565.
    We give classical proofs, strengthenings, and generalizations of Lecomte’s characterizations of analytic ω-dimensional hypergraphs with countable Borel chromatic number.
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  5.  16
    Measurable Chromatic Numbers.Benjamin D. Miller - 2008 - Journal of Symbolic Logic 73 (4):1139-1157.
    We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is N₀. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In (...)
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  6.  3
    Bases for Functions Beyond the First Baire Class.Raphaël Carroy & Benjamin D. Miller - 2020 - Journal of Symbolic Logic 85 (3):1289-1303.
    We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $\sigma $ -continuous with closed witnesses.
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  7.  5
    On the Existence of Small Antichains for Definable Quasi-Orders.Raphaël Carroy, Benjamin D. Miller & Zoltán Vidnyánszky - 2021 - Journal of Mathematical Logic 21 (2):2150005.
    We generalize Kada’s definable strengthening of Dilworth’s characterization of the class of quasi-orders admitting an antichain of a given finite cardinality.
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  8.  11
    Measurable Perfect Matchings for Acyclic Locally Countable Borel Graphs.Clinton T. Conley & Benjamin D. Miller - 2017 - Journal of Symbolic Logic 82 (1):258-271.
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  9.  6
    Recurrence and the Existence of Invariant Measures.Manuel J. Inselmann & Benjamin D. Miller - 2021 - Journal of Symbolic Logic 86 (1):60-76.
    We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.
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  10.  9
    A Generalization of the G0 Dichotomy and a Strengthening of the E^N0 Dichotomy.Benjamin D. Miller - forthcoming - Journal of Mathematical Logic.
    We generalize the ????0 dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi...
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  11.  3
    On the Existence of Large Antichains for Definable Quasi-Orders.Benjamin D. Miller & Zoltán Vidnyánszky - 2020 - Journal of Symbolic Logic 85 (1):103-108.
    We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.
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