11 problems

Let $\mathbb{S}$ be a Suslin ccc poset. A Suslin ccc poset is correctness-preserving if, given a correct diagram $\langle\mathbb{P}_0,\mathbb{P}_1,\mathbb{Q}_0,\mathbb{Q}_1\rangle$…

Let $A$ be a set and let $\rho$ be a partial order on $A$ with least element $0$ and greatest element $1$. Let $\operatorname{Pol}\rho$ be the clone of operations preserving $\rho$…

Let $A$ be a set and let $\rho$ be a partial order on $A$ with least element $0$ and greatest element $1$. Set $F=\operatorname{Pol}\rho$, the maximal clone of operations preservin…

Let $\mathsf{MA}(sigma\text{-centered})$ hold, and let $(P,<)$ be a partial order of size at most $2^{\mathfrak{c}}$ such that every $p\text{in}P$ has at most $\mathfrak{c}$ many p…

A partial order $(P,\leq_P)$ is locally countable if every element has at most countably many predecessors. The continuum is the cardinality of the real numbers, and the Turing deg…

Realization conjecture. If the order dimension of any locally countable partial order is $\mathfrak c$, then the cases besides $\mathrm{CH}$ are also realized; that is, there are l…

Let $k\geq 2$, and let $W_n$ be a set of $n$ random intervals. Write $E[k\operatorname{-wd}(W_n)]$ for the expected $k$-width of $W_n$. Random-interval set scaling conjecture. Ther…

Let $k\geq 2$, and let $R_n$ be a sequence of $n$ random intervals. Write $E[k\operatorname{-wd}(R_n)]$ for the expected $k$-width of $R_n$. Random-interval sequence scaling conjec…

Let $k\geq 2$ be an integer and set $\delta=1/k$. For a partial order $Q$ on $[n]$, let $e(Q)$ denote its number of linear extensions and let $\operatorname{comp}(Q)$ denote its nu…

Three sporadic entropic-order conjectures. The following three relations hold:

Let $G$ be the Hermitian Lie group considered above, with maximal bi-invariant order, maximal continuous bi-invariant order, and, when $G$ is of tube type, geometric order. Coincid…