Lecture 17 – Subset Lattices

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

1. Subset Lattices and their Properties

This video introduces subset lattices and cubes, and some of their properties. (5:09)

2. The Inductive Nature of Subset Lattices

Subset lattices can be built in an inductive manner. (1:47)

3. Hamiltonian Property for Subset Lattices

Via induction, we prove that subset lattices are Hamiltonian. (5:51)

4. The Width of Subset Lattices and Sperner’s Theorem

This video gives some facts regarding the width of subset lattices and states Sperner’s Theorem, which we prove in subsequent videos. (5:20)

5. Proof of Sperner’s Theorem (Part 1)

Here, we discuss why the number of maximal chains through a set A is |A|! (n – |A|)!. (8:22)

6. Proof of Sperner’s Theorem (Part 2)

What is the total number of maximal chains on a subset lattice? If we have two sets, A and B, and neither is a subset of the other, can a maximal chain go through both of them? (2:21)

7. Proof of Sperner’s Theorem (Combining Part 1 & 2)

This video completes our first proof of Sperner’s Theorem.  After the preparation in parts 1 and 2, the rest of the proof is just algebra! (10:36)

8. Ranked Posets and the Middle Level of a Ranked Poset

In order to give our second proof of Sperner’s Theorem, we need the concept of a ranked poset and its middle level. This video introduces these concepts and provides a few examples. (2:24)

9. A Symmetric Chain

This video introduces the concept of a symmetric chain, and provides some examples of them. (4:21)

10. Symmetric Chain Partitions

If we are given a ranked poset, does it have a symmetric chain partition? This video explores one example where we try to answer this question for a given ranked poset.

At 3:30, a student asks for clarification on partitions of posets into chains.  A partition into chains is a division of a poset into disjoint sets, where each of the sets is a chain.  It does not matter whether or not the sets are connected to each other in the poset diagram. (7:07)

11. Examples of Symmetric Chain Partitions

We want to show that the subset lattice has a symmetric chain partition. In this video, we state our goal, and discuss a few complicated examples of symmetric chain partitions for ranked posets of height 4 and of height 5. (3:07)

12. A Lemma: The Product of Two Chains has an SCP

To show that 2n has a symmetric chain partition, we need a lemma. The lemma is that the product of two chains has a symmetric chain partition. We describe what this lemma means and provide an example. At the end of the video we talk about products of longer chains. (5:50)

13. Finding Symmetric Chain Partitions in the First Few Subset Lattices

In this video we look at symmetric chain partitions of B1, B2, and B3. (5:05)

14. 2n Has a Symmetric Chain Partition

Here, we look at B, and see that by extending this process,  2n must have a symmetric chain partition for all n. (6:50)

15. Symmetric Chain Partitions & Dilworth’s Theorem

Going back to our symmetric chain partition for a ranked poset of height 5, we ask, what is the width? The symmetric chain partition helps us answer this question. (1:00)