8 problems

Consistency conjecture. It is consistent with ZF plus the Axiom of Dependent Choices (DC) that BPI holds and there is no Vitali set in the real line.

Let $W$ be the Cantor set $\mathscr{D}_{1}$ and let $R=\operatorname{Co}(W)$ be the countable atomless Boolean algebra. Let $I$ be the set of finite binary strings, including the e…

Let $B_n$ be the Boolean algebra of all subsets of $[n]$, ordered by inclusion. A chain decomposition of $B_n$ has a type, namely the partition obtained by rearranging its chain si…

Boolean-algebra decidability conjecture. The theory of Boolean algebras of cardinality less than $\lambda$, or the first-order theory of Boolean algebras expanded by quantification…

Let $\beta=\bigwedge_{n_0<\omega}\bigvee_{n_1<\omega}\cdots\omega^{k-1}\times A_{(n_0,n_1,\ldots,n_{k-1})}$ be the indicated iterated operation on a family of sets…

Boolean-ultrapower saturation conjecture. There exist such $cappa$, $cB$, $U$, $cM$, and $cN$ for which

Peckness conjecture. The edge poset of the quotient satisfies

The depth inequality conjecture. In ZFC,