A poset has an interval representation if each can be assigned a real interval so that in if and only if lies completely to the left of . Such orders are called interval orders. Fishburn (1983, 1985) proved that for any positive integer , an interval order has a representation in which all interval lengths are between and if and only if the order does not contain as an induced poset. In this paper, we give a simple proof of this result using a digraph model.
"A simple proof characterizing interval orders with interval lengths between 1 and $k$." Involve 11 (5) 893 - 900, 2018. https://doi.org/10.2140/involve.2018.11.893