[Paper Review] Ahlswede-Khachatrian Theorems: Weighted, Infinite, and Hamming

Yuval Filmus|arXiv (Cornell University)|Oct 3, 2016

Limits and Structures in Graph Theory10 references3 citations

TL;DR

This paper extends the Ahlswede–Khachatrian theorem to weighted, infinite, and Hamming settings by reworking the original proofs in the $μ_p$-measure framework. It fully characterizes the maximum $μ_p$-measure of $t$-intersecting families for all $p \in (0,1)$, including $p > 1/2$, and identifies all optimal families, resolving gaps in prior work that failed to describe optimal families or only applied for $p < 1/2$. The results unify and generalize classical extremal combinatorics across multiple domains.

ABSTRACT

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 $μ_p$ measure of a $t$-intersecting family on $n$ points, where $μ_p(A) = p^{|A|} (1-p)^{n-|A|}$. 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 $p &lt; 1/2$. 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 $p &gt; 1/2$, 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 $μ_p$ 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.

Motivation & Objective

• To complete the characterization of maximum $μ_p$-measure $t$-intersecting families on $n$ points, including for $p > 1/2$, where prior methods failed.
• To identify all optimal families in the weighted setting, resolving the lack of structural description in earlier reductions.
• To extend the Ahlswede–Khachatrian theorem to the case of infinitely many points, defining $w_{\infty}(p,t)$ as the supremum $μ_p$-measure of $t$-intersecting families in $\{0,1\}^{\aleph_0}$.
• To generalize the theorem to the Hamming scheme $\mathbb{Z}_m^n$, showing it corresponds to $\u03bc_p$ with $p = s/m$, and determine maximum families.
• To provide a simplified, unified proof of the Erdős–Ko–Rado theorem in the $μ_p$ setting using a generalized circle argument.

Proposed method

• Rephrasing the two original proofs of Ahlswede and Khachatrian (1997) in the weighted $μ_p$-measure setting to preserve structural information about optimal families.
• Using a lifting argument via random restrictions to transfer results from finite to infinite settings, showing that $w_{\infty}(p,t) = \sup_n w(n,t,p)$.
• Applying Hoeffding’s inequality and measure concentration to bound the $μ_p$-measure of families under $p$-biased sampling, enabling comparison with optimal families.
• Extending Katona’s circle method to the $μ_p$ setting to give a concise proof of the Erdős–Ko–Rado theorem in the weighted case.
• Defining and analyzing the family $\mathcal{F}_{t,r} = \{S : |S \cap [t+2r]| \geq t+r\}$ in the $μ_p$-measure framework to determine optimal thresholds for $p$.
• Using permutation invariance and minimality arguments to prove uniqueness of optimal families under $p$-optimality, especially when $p$ is not a critical fraction $r/(t+2r-1)$.

Experimental results

Research questions

• RQ1 What is the maximum $μ_p$-measure of a $t$-intersecting family on $n$ points for all $p \in (0,1)$, including $p > 1/2$?
• RQ2 Which families achieve this maximum measure, and how can they be fully characterized in the weighted setting?
• RQ3 What is the supremum $μ_p$-measure of a $t$-intersecting family on infinitely many points, and how does it relate to the finite case?
• RQ4 How does the Ahlswede–Khachatrian theorem extend to the Hamming scheme $\mathbb{Z}_m^n$, and what is the correspondence with $\u03bc_p$ for $p = s/m$?
• RQ5 Can Katona’s circle argument be generalized to the $μ_p$ setting to yield a simpler proof of the Erdős–Ko–Rado theorem?

Key findings

• The paper fully characterizes $w(n,t,p)$, the maximum $μ_p$-measure of a $t$-intersecting family on $n$ points, for all $n$, $t$, and $p \in (0,1)$, resolving the incomplete picture from prior work.
• For $p \in \left[\frac{r}{t+2r-1}, \frac{r+1}{t+2r+1}\right]$, the maximum measure is $\mu_p(\mathcal{F}_{t,r})$, and this family is optimal for all $n \geq t+2r$, with explicit identification of all optimal families.
• The case $p > 1/2$ is resolved using a different technique of Ahlswede and Khachatrian, showing that the optimal families remain of the form $\mathcal{F}_{t,r}$, but with $r$ chosen based on $p$.
• The infinite case yields $w_{\infty}(t,p) = \sup_n w(n,t,p)$, and the supremum is achieved in the limit, with optimal families lifted from finite constructions.
• In the Hamming scheme $\mathbb{Z}_m^n$, the maximum $t$-intersecting families correspond to $\u03bc_p$ with $p = s/m$, and the optimal families are again $\mathcal{F}_{t,r}$ for appropriate $r$, with explicit bounds on $n$.
• A generalized circle argument in the $μ_p$ setting provides a short, elegant proof of the Erdős–Ko–Rado theorem, simplifying prior proofs by Dinur–Friedgut and Friedgut.