A Simple Proof Characterizing Interval Orders with Interval Lengths between 1 and $k$

A poset $P=(X,\prec )$ 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 \emph{interval orders}. Fishburn 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 $(\mathbf{k}+\mathbf{2})+\mathbf{1}$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.