Michał Seweryn, Dimension and standard examples in planar posets

The dimension of a poset is the least integer $d$ such that the poset is isomorphic to a subposet of the product of $d$ linear orders. In 1983, Kelly constructed planar posets of arbitrarily large dimension. Crucially, the posets in his construction involve large standard examples, the canonical structure preventing a poset from having small dimension. Kelly’s construction inspired one of the most challenging questions in dimension theory: are large standard examples unavoidable in planar posets of large dimension? We answer the question affirmatively by proving that every $d$-dimensional planar poset contains a standard example of order $\Omega(d)$. More generally, we prove that every poset from Kelly’s construction appears in every poset with a planar cover graph of sufficiently large dimension.

joint work with Heather Smith Blake, Jędrzej Hodor, Piotr Micek, and William T. Trotter.