  The seminal complete intersection theorem of Ahlswede and Khachatrian gives
the maximum cardinality of a $k$-uniform $t$-intersecting family on $n$ points,
and describes all optimal families. We extend this theorem to several other
settings: the weighted case, the case of infinitely many points, and the
Hamming scheme.
  The weighted Ahlswede-Khachatrian theorem gives the maximal $\mu_p$ measure
of a $t$-intersecting family on $n$ points, where $\mu_p(A) = p^{|A|}
(1-p)^{n-|A|}$. As has been observed by Ahlswede and Khachatrian and by Dinur
and Safra, this theorem can be derived from the classical one by a simple
reduction. However, this reduction fails to identify the optimal families, and
only works for $p < 1/2$. We translate the two original proofs of Ahlswede and
Khachatrian to the weighted case, thus identifying the optimal families in all
cases. We also extend the theorem to the case $p > 1/2$, using a different
technique of Ahlswede and Khachatrian (the case $p = 1/2$ is Katona's
intersection theorem). We then extend the weighted Ahlswede-Khachatrian theorem
to the case of infinitely many points.
  The Ahlswede-Khachatrian theorem on the Hamming scheme gives the maximum
cardinality of a subset of $\mathbb{Z}_m^n$ in which any two elements $x,y$
have $t$ positions $i_1,\ldots,i_t$ such that $x_{i_j} - y_{i_j} \in
\{-(s-1),\ldots,s-1\}$. We show that this case corresponds to $\mu_p$ with $p =
s/m$, extending work of Ahlswede and Khachatrian, who considered the case $s =
1$. We also determine the maximum cardinality families. We obtain similar
results for subsets of $[0,1]^n$, though in this case we are not able to
identify all maximum cardinality families.
