Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Measurability and Perfect Set Theorems for Equivalence Relations with
  Small Classes   We ask whether $\mathbf{\Delta^1_2}$ or $\mathbf{\Sigma^1_2}$ equivalence
relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly
many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive
answer for $\mathbf{\Delta^1_2}$ equivalence relations is equivalent to the
$I$-measurability of $\mathbf{\Delta^1_2}$ sets. However, the analogous
statement for $\mathbf{\Sigma^1_2}$ equivalence relations is false:
$\mathbf{\Sigma^1_2}$ equivalence relations with meager classes have a perfect
set of pairwise inequivalent elements if and only if $\mathbf{\Delta^1_2}$ sets
have the Baire property.

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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: gr-qc