8 problems

Linear-size Borel independent complete sections conjecture. There exist constants $C,0>0$ with the following property: if the maximum degree of $G_1$ is at most $0$ and every compo…

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

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.

The hyperfiniteness conjecture. Every countable amenable Borel equivalence relation is hyperfinite.

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…