15 problems

Let $P$ be a finite poset. A local realizer of $P$ is a family of partial linear extensions whose frequency is the maximum number of partial linear extensions in which any element…

Standard-example–wheel conjecture. For every such poset $P$,

Multiplicative lower-bound conjecture. Among cover-planar posets,

Dimension-boundedness conjecture. The class of cover-planar posets is $\dim$-bounded.

Joret–Micek conjecture. The dimension of $P$ is bounded in terms of the number of minimal elements of $P$ and the treewidth of the cover graph.

Huynh's characterization conjecture. For minor-closed classes of graphs, the bounded-dimension property holds exactly for those classes that exclude the cover graph of a poset in K…

Let $H$ be a fixed graph, and let a poset's cover graph be $H$-minor-free if it contains no $H$ minor. A chain in a poset is a set of pairwise comparable elements; two chains are i…

Let $H$ be a graph. Call $H$ unavoidable if the cover graph of every poset of sufficiently large dimension contains $H$ as a minor. Kelly's construction refers to the family of pla…

Let $g$ be a non-negative integer, and consider graphs of genus $g$ and their adjacency posets $A_G$. Felsner, Li and Trotter's conjecture states that the dimension of $A_G$ should…

Let $H_d\simeq \mathbb{R}^{d-1}$ be the hyperplane used for the $d$-dimensional construction, let $R$ be a $d$-representation, and let $\Sigma(R)$ be the abstract simplicial comple…

A simplicial complex has Dushnik-Miller dimension at most $d$ if and only if it is a subcomplex of a TD-Delaunay complex in $\mathbb{R}^{d-1}$. Dushnik-Miller–TD-Delaunay conjectur…

Let $\mathcal{C}$ be a monotone class of bipartite graphs. Regard each graph in $\mathcal{C}$ as a poset of height at most $2$. Bounded-average-degree conjecture. If these posets h…

Let $\mathcal{C}$ be a monotone class of graphs. For each fixed $h\geq 1$, consider posets of height at most $h$ whose cover graphs belong to $\mathcal{C}$. Bounded-expansion chara…

Bounded-expansion characterization. The class $\mathcal{C}$ has bounded expansion if and only if, for each $h\geq 1$, the posets of height $h$ whose cover graphs are in…

Let $P$ be a poset, and let its cover graph be the graph whose vertices are the elements of $P$ and whose edges join pairs in a cover relation. A graph is planar if it can be drawn…