Planar Posets Have Dimension at Most Linear in Their Height

We prove that every planar poset P of height h has dimension at most 192h + 96. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets of height h and dimension at least (4/3)h− 2. (G. Joret) Computer Science Department, Université Libre de Bruxelles, Brussels, Belgium (P. Micek) Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland, and Institute of Mathematics, Combinatorics and Graph Theory Group, Freie Universität Berlin, Berlin, Germany (V. Wiechert) Institut für Mathematik, Technische Universität Berlin, Berlin, Germany E-mail addresses: [email protected], [email protected], [email protected]. Date: September 26, 2017.