= Ahlswede-Khachatrian theorem
{c}
{title2=$M(n,r,t)=\max_{0\leq i\leq r-t}|\mathcal A_i|$}
{wiki=Ahlswede–Khachatrian_theorem}
= Complete intersection theorem
{synonym}
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 $n\leq2r-t$, 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.
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