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

TL;DR

This paper provides a straightforward proof characterizing interval orders with interval lengths between 1 and k, based on the absence of a specific induced poset, using a digraph model.

Contribution

It offers a simple, new proof of Fishburn's characterization of interval orders with bounded interval lengths via a digraph approach.

Findings

Characterization of interval orders with interval lengths between 1 and k.

Proof uses a digraph model for simplicity.

Identifies forbidden induced poset (k+2)+1) for such interval orders.

Abstract

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 $\mathbf{(k+2)+1}$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.