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7. Miscellaneous

    1.     Call a finite set $F \subset \R^k$ Ramsey if for every finite $c$, there is a large enough $n \geq k$ such that any $c$-coloring of $\R^n$ yields a monochromatic copy of $F$. Call $F$ Borel Ramsey if the same holds where we only consider Borel colorings of $\R^n$. Call $F$ spherical if there is some $k-1$-sphere in $\R^k$ containing F.

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      Problem 7.1.

      1. Is every $F$ Borel Ramsey? We are especially interested in not spherical $F$.
      2. If $F$ is Borel Ramsey, is it Ramsey?
          It is known that non-spherical $F$ are not Ramsey


        • Add remark

          Problem 7.2.

          Among all Polish $\Q$-vector spaces ordered by continuous $\Q$-linear injectability, is there anything between $\ell^1(\Q)$ and $\R$?


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              Problem 7.3.

              Let $(X,\mu),(Y,\nu)$ be standard probability spaces with $\mu$ and $\nu$ atomless. Let $T$ and $G$ be Borel pmp graphs on $(X,\mu)$ and $(Y,\nu)$ resp. such that
              1. $T$ is acyclic and has bounded degree.
              2. For some $\epsilon > 0$, For all $y \in Y$, $\nu(N_G(y)) > \frac{1}{2} + \epsilon$.
              Is there a measure preserving embedding from $T$ to $G$?
                  **** I have specifically been told that this problem has a trivial negative answer as stated and will need to be amended.


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                  Problem 7.4.

                  Let $f$ be a countable-to-1 aperiodic (i.e. acyclic) Borel function which is well founded as a binary relation (no infinite back-orbits).
                  1. Is there a Borel complete section (for $f$’s orbit equivalence relation) $B$ such that $f[B] \subset B$ and $f$ restricted to $B$ is finite-to-1? (There is a $\mu$-measurable such $B$ for a given Borel probability measure $\mu$ on the space.)
                  2. What if we only require $B$ to be Baire measurable?
                      Resolved during the workshop with negative answer even to part (b).


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                      Problem 7.5.

                      Is the measurable arboricity of a Borel graph always equal to its actual arboricity?
                          This false for Borel arboricity. Some bounds in nice cases were achieved during the workshop.

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