 Ahlswede-Khachatrian theorem

The maximum size of a t-intersecting family of $r$-sets on $[n]$ is the largest complete-intersection candidate family. In strict nontrivial intervals $n_{k+1}<n<n_k$, the maximizing index is $k$, where $n_k=(r-t+1)(2+(t-1)/k)$ and $n_0=\infty$. At admissible interior boundaries the consecutive candidates tie. If $n\le 2r-t$, the whole level is optimal. A proof uses left compression, the small-support generating lemma for extremal intersecting families for the original and complement families, and the compatibility bound for complementary generating families. The latter forces one generator family to have the candidate threshold size, placing the original family inside a candidate.