# Lecture 14 – Posets: Mirsky’s & Dilworth’s Theorems

You may want to download the lecture slides that were used for these videos (PDF).

## 1. Terminology Review

Here, we review cover graphs, poset diagrams, comparability graphs, and incomparability graphs. (5:19)

## 2. Example: a Poset on 26 Points

This video gives a set of statements regarding a poset. Do you agree with all of the statements? (6:48)

## 3. Chains, Maximal Chains, and Chain Height

In this video we discuss a number of concepts: chains, maximal chains, and chain heights. (5:48)

## 4. Antichains and Maximal Antichains

Definitions and examples of antichains and maximial antichains are given. (2:50)

## 5. Width, and Partitioning Posets

The width of a poset is the maximum size of an antichain in P.  If a poset can be partitioned into t antichains, height(P) ≤ t.  Similarly, if a poset can be partitioned into t chains, width(P) ≤ t.  To find a partition into antichains, we can color a poset.  (9:24)

## 6. A Partition Shows Width ≤ 9

Elements in this graph that have the same coloring form a chain. Can you see why? (5:09)

## 7. Mirsky’s Theorem (Dual to Dilworth’s Theorem)

A poset of height h can be partitioned into h antichains.  The proof here also provides an algorithm to find the height and a partition into h antichains.  (12:55)

## 8. Dilworth’s Theorem

A poset of width w can be partitioned in to w chains. Despite how similar this statement sounds to Mirsky’s Theorem, the proof of this theorem is much harder.  (5:14)

## 9. The Proof of Dilworth’s Theorem (1)

Our proof of Dilworth’s Theorem is divided into three parts. This video provides the first part of the proof. (5:12)

## 10. The Proof of Dilworth’s Theorem (2)

Our proof of Dilworth’s Theorem is divided into three parts. This video provides the second part of the proof. (5:29)

## 11. The Proof of Dilworth’s Theorem (3)

Our proof of Dilworth’s Theorem is divided into three parts. This video provides the last part of the proof. (4:59)

## 12. Historical Notes

This video provides some historical background to the development to Dilworth’s Theorem. (5:38)