A simple proof characterizing interval orders with interval lengths between 1 and $k$

Abstract

A poset $P=\left(X,\prec \right)$ has an interval representation if each $x\in X$ can be assigned a real interval $I_x$ so that $x\prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called interval orders. Fishburn (1983, 1985) proved that for any positive integer $k$, an interval order has a representation in which all interval lengths are between $1$ and $k$ if and only if the order does not contain $\left(k+2\right)+1$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.