Typical intersecting families at $n=2k+1$ and $n=2k+2$

A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of $k$-uniform intersecting families on $2k+1$ and $2k+2$ elements as $k\to\infty$. For $n=2k+2$, we prove that almost all intersecting familie