Monotone paths in ordered graphs

LetVfin andEfin resp. denote the classes of graphsG with the property that no matter how we label the vertices (edges, resp.) ofG by members of a linearly ordered set, there will exist paths of arbitrary finite lengths with monotonically increasing labels. The classesVinf andEinf are defined similarly by requiring the existence of an infinite path with increasing labels. We proveEinf ⫋Vinf ⫋Vfin ⫋Efin. Finally we consider labellings by positive integers and characterize the class corresponding toVinf.