9 problems

Nonadjacent maximum-degree measurable edge-coloring conjecture. For every probability measure $\mu$ on $V$,

Finite-degree Borel Shannon edge-coloring conjecture. If $\mathsf{asi}(G)<\infty$, then

Type $(4,3)$ finite asymptotic separation index matching conjecture. If $\mathsf{asi}(G)<\infty$, then $G$ has a Borel matching covering $A$.

Finite asymptotic separation index matching conjecture. If the bipartition has combinatorial expansion and $\mathsf{asi}(G)<\infty$, then $G$ has a Borel matching covering $A$.

Borel CSP dichotomy conjecture. For every $\mathcal{D}$, the set $\mathrm{CSP}_B(\mathcal{D})$ is either in $\mathbf{\Pi}^{1}_{1}$ or is $\mathbf{\Sigma}^{1}_{2}$-complete.

Let $P=(V,\preceq)$ be a Borel poset. Its height is the largest size of a chain in $P$, and an antichain is a set of pairwise incomparable elements. Borel dual Dilworth conjecture.…

Let $P=(V,\preceq)$ be a Borel poset, meaning that $V$ is a standard Borel space and the comparability relation is a Borel subset of $V\times V$. Its width is the maximum size of a…

Torsion-free marked groups conjecture. For every $k\geq 3$, there are such marked groups $\Gamma_k$ and $\Delta_k$ with each group torsion free.

Let $X$ be a standard Borel space, and let a finitely generated amenable group act freely on $X$ by Borel isomorphisms. The associated Borel graph has an edge between distinct poin…