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The seminal complete intersection theorem of Ahlswede and Khachatrian givesthe maximum cardinality of a $k$-uniform $t$-intersecting family on $n$ points,and describes all optimal families. We extend this theorem to several othersettings: the weighted case, the case of infinitely many points, and theHamming scheme.  The weighted Ahlswede-Khachatrian theorem gives the maximal $\mu_p$ measureof 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 Dinurand Safra, this theorem can be derived from the classical one by a simplereduction. However, this reduction fails to identify the optimal families, andonly works for $p < 1/2$. We translate the two original proofs of Ahlswede andKhachatrian to the weighted case, thus identifying the optimal families in allcases. We also extend the theorem to the case $p > 1/2$, using a differenttechnique of Ahlswede and Khachatrian (the case $p = 1/2$ is Katona'sintersection theorem). We then extend the weighted Ahlswede-Khachatrian theoremto the case of infinitely many points.  The Ahlswede-Khachatrian theorem on the Hamming scheme gives the maximumcardinality 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 similarresults for subsets of $[0,1]^n$, though in this case we are not able toidentify all maximum cardinality families.

Ahlswede-Khachatrian Theorems: Weighted, Infinite, and Hamming