Prove that the only antichain of S={1,2,3,4} of size 6 is the antichain of all 2-subsets of S.

Name: Problem 30, Chapter 5: Introductory Combinatorics
Uploaded: 2025-07-05T22:20:52-08:00
Duration: 1 min 30 s
Channel: Nick Johnson
Description: Prove that the only antichain of S={1,2,3,4} of size 6 is the antichain of all 2-subsets of S.

step 1 All right, hello. So for Part A, we let here S be in our set S, and then we have F be an element

Prove that the only antichain of S={1,2,3,4} of size 6 is...

Key Concepts

Sperner's Theorem

Sperner's Theorem states that the largest antichain in the power set of an n-element set is given by the largest level, which consists of all subsets of size floor(n/2) (or ceiling(n/2), which is equal when n is even). This theorem is pivotal in understanding the extremal properties of antichains in a subset lattice.

Level Sets

Within the subset lattice, level sets are the collections of subsets that have the same number of elements. All subsets at the same level are incomparable under inclusion, making each level set an antichain. This concept is crucial when determining maximal antichains in the lattice.

Binomial Coefficient

The binomial coefficient, often denoted as C(n, k) or 'n choose k', represents the number of k-element subsets of an n-element set. It quantifies the size of each level set in the subset lattice, which is key to identifying the largest possible antichain as described by Sperner's Theorem.

Antichain

An antichain in a partially ordered set (poset) is a collection of elements such that no one element of the collection is related to any other under the poset’s order relation. This means that in an antichain, for any two distinct elements, neither is a subset (or is less than) the other.

Subset Lattice

The power set of a set, ordered by inclusion, forms a lattice known as the subset lattice. In this lattice, every element is a subset, and the ordering is given by the inclusion relation. This structure is fundamental in combinatorics, as it organizes subsets into levels based on their size.

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