# 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)