12 problems

A standard example of dimension $m$ is the poset formed by the one-element subsets and the $(m-1)$-element subsets of an $m$-element set, ordered by inclusion. Standard-example con…

Let $P$ and $Q$ be finite nonempty posets, and let $\dim(P)$ denote the order dimension of $P$. Product-dimension conjecture. … This is described as a longstanding conjecture about…

Let $P$ and $Q$ be posets. Write $P\times Q$ for their Cartesian product, ordered componentwise, and let $\dim(P)$ denote the least integer $n$ such that $P$ is isomorphic to a sub…

Bubble-lattice order-dimension conjecture. For all $m,n\geq 0$,

Let $\mathcal{D}_{[n]}$ be the divisibility poset on $[n]$. Write $\dim(\mathcal{D}_{[n]})$ for its order dimension, and let $\dim_2(\mathcal{D}_{[n]})$ be its 2-dimension, the sma…

Let $(X,\leq)$ be an ordered set, and let $\operatorname{tig}(X,\leq)$ denote its transitive incomparability graph. A two-dimensional extension of $(X,\leq)$ is an ordered set obta…

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 $P$ be a poset with at most $2n+1$ points, and let $obreaktext{dimension}(P)$ denote its dimension. For a positive integer $d$, let $Sd$ be the standard example of size $d$. *…

Trotter's Removable Pair Conjecture. Every finite poset on at least $3$ points has a removable pair.

Let $P$ be a finite poset, let $|P|$ denote its number of elements, and let $\dim(P)$ denote its order dimension. For a positive integer $m$, let $S_m$ be the standard example on…

Let $P$ be a poset, let $D$ be its order diagram, let $E$ be the edge set of $D$, and let $\phi:E\to\{1,2,\dots,r\}$ be an $r$-coloring of the edges. For a subdiagram $D'$ with ass…

Reversal-ratio conjecture.