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We describe some metric properties of incomparability graphs. We consider theproblem of the existence of infinite paths, either induced or isometric, in theincomparability graph of a poset. Among other things, we show that if theincomparability graph of a poset is connected and has infinite diameter, thenit contains an infinite induced path. Furthermore, if the diameter of the setof vertices of degree at least $3$ is infinite, then the graph contains as aninduced subgraph either a comb or a kite.

Metric properties of incomparability graphs with an emphasis on paths