Dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if P has dimension d, then to know whether x ≤ y in P it is enough to check whether x ≤ y in each of the d linear extensions of a witnessing realizer. Focusing on the encoding aspect, Nešetřil and Pudlák defined a more expressive version of dimension. A poset P has Boolean dimension at most d if it is possible to decide whether x ≤ y in P by looking at the relative position of x and y in only d linear orders on the elements of P (not necessarilly linear extensions). We prove that posets with cover graphs of bounded tree-width have bounded Boolean dimension. This stands in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. This result might be a step towards a resolution of the long-standing open problem: Do planar posets have bounded Boolean dimension?
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Stefan Felsner and Piotr Micek were partially supported by DFG grant FE-340/11-1.
Piotr Micek was partially supported by the National Science Center of Poland, grant no. 2015/18/E/ST6/00299.
Tamás Mészáros was supported by the Dahlem Research School of Freie Universität Berlin.
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Felsner, S., Mészáros, T. & Micek, P. Boolean Dimension and Tree-Width. Combinatorica 40, 655–677 (2020). https://doi.org/10.1007/s00493-020-4000-9
Mathematics Subject Classification (2010)