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5. Graphings, Treeings, and CBERs (countable Borel equivalence relations)

Let $X$ be a Polish space, $E$ and $F$ CBERs on $X$, and $\mu$ a Borel probability measure on $X$.


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      Problem 5.1.

      Let $E$ be treeable.
      1. Does every Borel graphing, say, $G$ of $E$ contain a Borel subgraph, say $T$, which is a treeing of $E$?
      2. What if we just want $T$ to be $\mu$-measurable?


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          Problem 5.2.

          Let $E$ be pmp and $G$ a Borel graphing of $E$. Does $G$ contain Borel subgraphings (of $E$) whose cost is arbitrarily close to the cost of $E$?


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              Problem 5.3.

              Is every finite index extension of a hypersmooth equivalence relation hypersmooth?


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                  Problem 5.4.

                  Let $E$ be treeable.
                  1. Is every finite index extension of E treeable?
                  2. What if we only require a $\mu$-measurable treeing.


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                      Problem 5.5.

                      Suppose $E$ and $F$ are treeable of type III_A. do they have class-bijective lifts which are stably orbit equivalent?


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                          Problem 5.6.

                          Is there an “informative” notion of cost in the quasi-pmp setting?

                              Cite this as: AimPL: Descriptive graph theory, available at http://aimpl.org/descriptgraph.