## Involve: A Journal of Mathematics

• Involve
• Volume 11, Number 5 (2018), 893-900.

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

#### Abstract

A poset $P=(X,≺)$ has an interval representation if each $x∈X$ can be assigned a real interval $Ix$ so that $x≺y$ in $P$ if and only if $Ix$ lies completely to the left of $Iy$. 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 $(k+2)+1$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.

#### Article information

Source
Involve, Volume 11, Number 5 (2018), 893-900.

Dates
Revised: 30 January 2018
Accepted: 5 February 2018
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.involve/1523498552

Digital Object Identifier
doi:10.2140/involve.2018.11.893

Mathematical Reviews number (MathSciNet)
MR3784035

Zentralblatt MATH identifier
1385.05057

#### Citation

Boyadzhiyska, Simona; Isaak, Garth; Trenk, Ann N. A simple proof characterizing interval orders with interval lengths between 1 and $k$. Involve 11 (2018), no. 5, 893--900. doi:10.2140/involve.2018.11.893. https://projecteuclid.org/euclid.involve/1523498552

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