Theorem bounding the size of families of intersecting sets in layers of the Power set

For $r \leq n/2$, if $\mathcal{A} \subseteq [n]^{(r)}$ is intersecting then $|\mathcal{A}| \leq \binom{n-1}{r-1}$. For $r < n/2$, the maximum families are all of the form {all sets which have at least a certain element $a$ in common}. For $r=n/2$, there are more possible types of maximum families.

See here for proofs.

https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem

An intersecting family of r-element sets may be maximal, in that no further set can be added without destroying the intersection property, but not of maximum size. An example with n = 7 and r = 3 is the set of 7 lines of the Fano plane, much less than the Erdős–Ko–Rado bound of 15.

Applications

Two families theorem

Generalization of theorem