---
tag: RA5SQ908G1
kind: concept
name: Width of a poset
msc: [06A07]
sources:
  - url: https://en.wikipedia.org/wiki/Dilworth's_theorem
---

## Summary

The width of a [[partially ordered set|0QV2JS9RJ3]] is the largest size of an [[antichain|03HF498P1J]] in it.
It measures how many elements can be pairwise incomparable at the same time.
By [[Dilworth's theorem|GT224XDA4E]], for a finite poset it is also the least number of chains that cover the poset.

## Explanations

### L1 human

Draw a poset with smaller elements lower down.
The width is the largest number of elements that sit side by side, with no two of them related.
A chain has width $1$.
In the subsets of $\set{1, 2, 3}$ ordered by inclusion, the three one-element subsets form an antichain, and no antichain is larger, so the width is $3$.