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

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)