The dim-boundedness conjectures for planar posets

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 vertices are the elements of the poset and whose edges join pairs in a cover relation. The dimension of a poset $P$, denoted $\dim(P)$, is the least number of linear extensions whose intersection is $P$. A class of posets is dim-bounded if the dimensions of its members are bounded by a function of the relevant structural parameter—in particular, here by an absolute constant for the stated class. The planar-poset dim-boundedness conjectures. (1) The class of posets with planar diagrams is dim-bounded. (2) The class of posets with planar cover graphs is dim-bounded. These statements address whether planarity, despite allowing posets of arbitrarily large dimension in known constructions, forces dimension to remain bounded. The paper discusses the longstanding problem and notes that the planar cover-graph case is pursued through results forcing large standard examples; the supplied text does not establish either assertion.