Double counting
This part is devoted to the idea that, in order to prove a theorem about a complicated structure, it is sometimes enough to prove a statement about a simple substructure and then average it over different choices of that substructure.
Summary: the Erdos-Ko-Rado theorem, Sperner theorem, Bollobas’ set-pair inequality.
For bigger picture see Extremal Set Theory Course
A family $\mathcal F$ is intersecting if $F_1\cap F_2\ne \emptyset$ for any two sets from $\mathcal F$. The Erdos-Ko-Rado theorem gives a sharp upper bound on the size of an intersecting $\mathcal F\subset {[n]\choose k}$.
The proof we are going to learn is via Katona’s circle method. Here is the video by Tim Gowers:
Alternatively, here is the video by Gyula Katona himself: the first 10 minutes of the video
A family $\mathcal F$ is an antichain if there are no $F_1,F_2\in \mathcal F$ such that $F_1\subsetneq F_2$. Sperner's theorem gives the largest size of an antichain in $2^{[n]}$.
We are going to see the proof of Sperner’s theorem using chains. Here is the video by Tim Gowers:
Alternatively, here is the video by Gyula Katona.
The statement is on minute 6. He gives several proofs of Sperner’s theorem, the first one [minutes 8-32] is based on biregular bipartite graphs and uses the same ideas as shown at the end of the Tim Gowers video on double counting in the Preliminaries section. The second proof is using chains [minutes 32-42].
Bollobás’ set-pair inequality is a very useful generalization of Sperner's theorem. Take a look at pages 4-5 of the book by Gerbner and Patkós “Extremal Finite Set Theory”.