Weighted version

Katona's intersection theorem^[10] determines the maximum size of an intersecting family of subsets of ${\displaystyle \{1,\dots ,n\}}$. When ${\displaystyle {1}}$ is odd, the unique optimal family consists of all sets of size at least ${\displaystyle m+1}$ (corresponding to the Majority function), and when ${\displaystyle {1}}$ is odd, the unique optimal families consist of all sets whose intersection with a fixed set of size ${\displaystyle 2m-1}$ is at least ${\displaystyle m-1}$ (Majority on ${\displaystyle 2m-1}$ coordinates).

Friedgut^[11] considered a measure-theoretic generalization of Katona's theorem, in which instead of maximizing the size of the intersecting family, we maximize its ${\displaystyle \mu _{p}}$-measure, where ${\displaystyle \mu _{p}}$ is given by the formula

${\displaystyle \mu _{p}(S)=p^{|S|}(1-p)^{n-|S|}.}$

The measure ${\displaystyle \mu _{p}}$ corresponds to the process which chooses a random subset of ${\displaystyle \{1,\dots ,n\}}$ by adding each element with probability p independently.

Katona's intersection theorem is the case ${\displaystyle {1}}$. Friedgut considered the case ${\displaystyle p<1/2}$. The weighted analog of the Erdős–Ko–Rado theorem^[12] states that if ${\displaystyle {\cal {F}}}$ is intersecting then ${\displaystyle \mu _{p}({\cal {F)\leq p}}}$ for all ${\displaystyle p<1/2}$, with equality if and only if ${\displaystyle {\cal {F}}}$ consists of all sets containing a fixed point. Friedgut proved the analog of Wilson's result^[13] in this setting: if ${\displaystyle {\cal {F}}}$ is t-intersecting then ${\displaystyle \mu _{p}({\cal {F)\leq p^{t}}}}$ for all ${\displaystyle p<1/(t+1)}$, with equality if and only if ${\displaystyle {\cal {F}}}$ consists of all sets containing t fixed points. Friedgut's techniques are similar to Wilson's.

Dinur and Safra^[14] and Ahlswede and Khachatrian^[15] observed that the Ahlswede–Khachatrian theorem implies its own weighted version, for all ${\displaystyle p<1/2}$. To state the weighted version, we first define the analogs of the Frankl families:

${\displaystyle {\mathcal {F}}_{n,t,r}=\{A\subseteq \{1,\dots ,n\}:|A\cap \{1,\dots ,t+2r\}|\geq t+r\}.}$

The weighted Ahlswede–Khachatrian theorem states that if ${\displaystyle {\cal {F}}}$ is t-intersecting then for all ${\displaystyle 0<p<1}$,

${\displaystyle \mu _{p}({\mathcal {F}})\leq \max _{r\colon t+2r\leq n}\mu _{p}({\mathcal {F}}_{n,t,r}),}$

with equality only if ${\displaystyle {\cal {F}}}$ is equivalent to a Frankl family. Explicitly, ${\displaystyle {\mathcal {F}}_{n,t,r}}$ is optimal in the range

${\displaystyle {\frac {r}{t+2r-1}}\leq p\leq {\frac {r+1}{t+2r+1}}.}$

The argument of Dinur and Safra proves this result for all ${\displaystyle p<1/2}$, without the characterization of the optimal cases. The main idea is that if we take a random subset of ${\displaystyle \{1,\dots ,N\}}$ of size ${\displaystyle \lfloor pN\rfloor }$, then the distribution of its intersection with ${\displaystyle \{1,\ldots ,n\}}$ tends to ${\displaystyle \mu _{p}}$ as ${\displaystyle N\to \infty }$.

Filmus^[16] weighted Ahlswede–Khachatrian theorem for all ${\displaystyle n,t,p}$ using the original arguments of Ahlswede and Khachatrian^[17][18] for ${\displaystyle p<1/2}$, and using a different argument of Ahlswede and Khachatrian, originally used to provide an alternative proof of Katona's theorem, for ${\displaystyle p\geq 1/2}$.^[19]

Version for strings

Ahlswede and Khachatrian proved a version of the Ahlswede–Khachatrian theorem for strings over a finite alphabet.^[20] Given a finite alphabet ${\displaystyle \Sigma }$, a collection of strings of length n is t-intersecting if any two strings in the collection agree in at least t places. The analogs of the Frankl family in this setting are

${\displaystyle {\mathcal {F}}_{n,t,r}=\{w\in \Sigma ^{n}:|w\cap w_{0}|\geq t+r\},}$

where ${\displaystyle w_{0}\in \Sigma ^{n}}$ is an arbitrary word, and ${\displaystyle |w\cap w_{0}|}$ is the number of positions in which w and ${\displaystyle w_{0}}$ agree.

The Ahlswede–Khachatrian theorem for strings states that if ${\displaystyle {\cal {F}}}$ is t-intersecting then

${\displaystyle |{\mathcal {F}}|\leq \max _{r\colon t+2r\leq n}|{\mathcal {F}}_{n,t,r}|,}$

with equality if and only if ${\displaystyle {\cal {F}}}$ is equivalent to a Frankl family.

The theorem is proved by a reduction to the weighted Ahlswede–Khachatrian theorem, with ${\displaystyle p=1/|\Sigma |}$.

Notes

Works cited