Ahlswede-Khachatrian Theorems: Weighted, Infinite, and Hamming

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 $p<1\mathrm{/}2$. 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 $p>1\mathrm{/}2$, using a different technique of Ahlswede and Khachatrian (the case $p=1\mathrm{/}2$ 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 $i_1,\dots ,i_t$ such that $x_{i_j} - y_{i_j} \in {-(s-1),\ldots,s-1}$\mathrm{.}Weshowthatthiscasecorrespondsto$\mu_p$with$p = s/m$,extendingworkofAhlswedeandKhachatrian,whoconsideredthecase$s = 1$. 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.