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