Define a poset with on the set with if

The claim is that there exists a chain of length among the set This will then imply there exists a chain of length among the set

Lemma: for
Proof: Induction on

Now, by Mirsky's theorem, the length of the longest chain is the minimum number of anti-chains that partition But Since note that we can define anti-chains such that consists of all elements for which the largest power of dividing it is The conclusion follows.

It feels really good as the material seems much easier compared to 5 months ago. I've done quite a few problems on posets already, so tomorrow time to knock out some problems with the theorems of Cauchy-Davenport, Schur, and Van der Waerden!