Journal Article10.1007/S11083-004-3345-9
Finite Paths are Universal
Jan Hubička,Jaroslav Nešetřil +1 more
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TL;DR: It is proved that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them, which implies that finite partially ordered sets are on-line representable by paths and their homomorphisms.
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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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Peter J. Cameron
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TL;DR: In this article, it was shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2] and a classification of these partially-ordered sets is given.
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