Planarity and dimension

Jędrzej Hodor (Jagiellonian University)

Dimension is arguably the best-known measure of complexity of posets (partially ordered sets). The dimension of a poset is the least integer d such that the poset is a subposet of the d-dimensional Euclidean space equipped with the product order. A canonical structure that forces dimension up is called a standard example. However, there are families of posets with large dimension and no standard examples. A long-standing open problem was determining if large dimension forces large standard examples in the case of planar posets. We confirm this statement. The goal of this talk is to introduce all the concepts necessary to understand this result and to discuss some related problems.

This is joint work with Heather Blake, Piotr Micek, Michał Seweryn, and William T. Trotter.

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