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Finite Paths are Universal

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Abstract

We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to new on-line dimensions.

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Authors and Affiliations

  1. Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, 11800, Praha, Czechia

    J. Hubička & J. Nešetřil

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  1. J. Hubička
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  2. J. Nešetřil
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Correspondence to J. Hubička.

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Mathematics Subject Classifications (2000)

06A06, 06A07, 05E99, 05C99.

J. Nešetřil: Supported by a Grant LN00A56 of the Czech Ministry of Education. The first author was partially supported by EU network COMBSTRU at UPC Barcelona.

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Hubička, J., Nešetřil, J. Finite Paths are Universal. Order 21, 181–200 (2004). https://doi.org/10.1007/s11083-004-3345-9

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  • DOI: https://doi.org/10.1007/s11083-004-3345-9

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