16 problems

A poset has a planar diagram if its cover graph can be drawn in the plane with every cover relation represented by a curve directed upwards. Its cover graph is the graph whose vert…

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…

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…

For positive integers $m$ and $n$, let $L(m,n)$ be the poset of $m$-tuples … ordered coordinatewise, with rank $\sum_{r=1}^m a_r$. A chain is saturated if it skips no rank, and sym…

Let $K_t$ denote the complete graph on $t$ vertices, and let a graph exclude $K_t$ as a minor if it has no graph minor isomorphic to $K_t$. The dimension of a poset $P$, denoted…

For positive integers $n_1,\ldots,n_r$, let $\mathbf{n_i}$ denote a chain of length $n_i$, and let a lattice be nice when every stable-partition type dominates only types that are…

Let $P$ be a poset, and let $\mathop{}\\!\operatorname{extsat}(n,P)$ denote the minimum size of an external $P$-saturated family for the Boolean lattice ${\mathcal B}_n$, as define…

For a finite poset $P$, let an external $P$-saturated family for the Boolean lattice ${\mathcal B}_n$ be a family as defined in the paper, and let…

A poset is Gorenstein when it is Cohen--Macaulay and Eulerian. Its $cd$-index is denoted by $\Phi_P(c,d)$; coefficientwise comparison means comparison of the coefficients of corres…

Let $C$ be a finite Coxeter group, equipped with its weak order. A ranked poset is strongly Sperner if, for every positive integer $r$, the maximum size of a union of $r$ antichain…

Almost-tiling conjecture. There exists a constant $c=c(P)$ such that for every positive integer $n$, there is a set $S\subseteq 2^{[n]}$ with $|S|\leq c$ such that…

Brightwell–Scheinerman's fractional Removable Pair Conjecture. Every finite poset has a pair of points whose removal decreases the fractional dimension by at most $1$.

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…

Consecutive-layer conjecture. For any fixed poset $P$,