[Paper Review] A Proof of the Deza-Frankl Conjecture

David Ellis|arXiv (Cornell University)|Jul 19, 2008

Limits and Structures in Graph Theory3 references1 citations

TL;DR

This paper proves the Deza-Frankl conjecture by establishing an exact stability result for t-intersecting families of permutations in $S_n$: for sufficiently large $n$, any such family not contained in a $t$-coset has size at most $(1 - 1/e + o(1))(n-t)!$, with equality only for 'double translates' of a specific extremal family. The result extends to the alternating group $A_n$.

ABSTRACT

A family of permutations (\mathcal{A} \subset S_{n}) is said to be (t)- extit{intersecting} if any two permutations in (\mathcal{A}) agree on at least (t) points, i.e. for any (\sigma, \pi \in \mathcal{A}), (|\{i \in [n]: \sigma(i)=\pi(i)\}| \geq t). It was recently proved by Friedgut, Pilpel and the author that for (n) sufficiently large depending on (t), a (t)-intersecting family (\mathcal{A} \subset S_{n}) has size at most ((n-t)!), with equality only if (\mathcal{A}) is a coset of the stabilizer of (t) points (or `(t)-coset' for short), proving a conjecture of Deza and Frankl. Here, we first obtain a rough stability result for (t)-intersecting families of permutations, namely that for any (t \in \mathbb{N}) and any positive constant (c), if (\mathcal{A} \subset S_{n}) is a (t)-intersecting family of permutations of size at least (c(n-t)!), then there exists a (t)-coset containing all but at most a (O(1/n))-fraction of (\mathcal{A}). We use this to prove an exact stability result: for (n) sufficiently large depending on (t), if (\mathcal{A} \subset S_{n}) is a (t)-intersecting family which is not contained within a (t)-coset, then (\mathcal{A}) is at most as large as the family \mathcal{D} & = & \{\sigma \in S_{n}: \sigma(i)=i \forall i \leq t, \sigma(j)=j extrm{for some} j > t+1\} && \cup \{(1 t+1),(2 t+1),...,(t t+1)\} which has size ((1-1/e+o(1))(n-t)!). Moreover, if (\mathcal{A}) is the same size as (\mathcal{D}) then it must be a `double translate' of (\mathcal{D}), meaning that there exist (\pi, au \in S_{n}) such that (\mathcal{A}=\pi \mathcal{D} au). We also obtain an analogous result for (t)-intersecting families in the alternating group (A_{n}).

Motivation & Objective

• Establish an exact stability result for $t$-intersecting families of permutations in $S_n$, extending the Deza-Frankl conjecture.
• Characterize the maximum possible size of $t$-intersecting families not contained in a $t$-coset.
• Prove that such extremal families must be 'double translates' of a canonical construction.
• Extend the result to $t$-intersecting families in the alternating group $A_n$.
• Provide a quantitative bound on the deviation from the $t$-coset structure for large $n$.

Proposed method

• Use a rough stability result to show that any $t$-intersecting family of size at least $c(n-t)!$ is mostly contained within a $t$-coset, with error decaying as $O(1/n)$.
• Apply spectral and combinatorial techniques to analyze the structure of permutation families avoiding the $t$-coset configuration.
• Construct the extremal family $\mathcal{D}$ as the union of the pointwise stabilizer of $[t]$ and a set of transpositions involving $t+1$.
• Prove that any $t$-intersecting family not in a $t$-coset cannot exceed the size of $\mathcal{D}$, which is $(1 - 1/e + o(1))(n-t)!$.
• Use group action invariance to define 'double translates' and show that equality in size implies such a structure.
• Adapt the method to the alternating group $A_n$ by restricting to even permutations and analyzing the corresponding stabilizer structure.

Experimental results

Research questions

• RQ1 What is the maximum size of a $t$-intersecting family of permutations in $S_n$ that is not contained in a $t$-coset?
• RQ2 How close can such a family be to the $t$-coset bound in terms of size?
• RQ3 What structural form must a $t$-intersecting family of maximum size (short of the $t$-coset) take?
• RQ4 Can the extremal structure be characterized up to group action, and if so, what is the precise form?
• RQ5 Does a similar stability and extremal structure result hold in the alternating group $A_n$?

Key findings

• For $n$ sufficiently large depending on $t$, any $t$-intersecting family in $S_n$ not contained in a $t$-coset has size at most $(1 - 1/e + o(1))(n-t)!$.
• The extremal family $\mathcal{D}$, defined as the union of the stabilizer of $[t]$ and transpositions $(i, t+1)$ for $i \leq t$, achieves this bound.
• If a $t$-intersecting family $\mathcal{A} \subset S_n$ has the same size as $\mathcal{D}$, then $\mathcal{A}$ must be a 'double translate' of $\mathcal{D}$, i.e. $\mathcal{A} = \pi \mathcal{D} \tau$ for some $\pi, \tau \in S_n$.
• A rough stability result shows that any $t$-intersecting family of size at least $c(n-t)!$ is contained in a $t$-coset up to an $O(1/n)$-fraction of elements.
• The same extremal bound and structural characterization hold for $t$-intersecting families in the alternating group $A_n$.
• The result confirms the Deza-Frankl conjecture in its exact form, with a precise characterization of extremal families beyond the $t$-coset case.