A Complete Intersection Theorem for Large Permutation Groups

2026-06-30

AI Summary

This work resolves the long-standing problem posed by Deza and Frankl in 1977 concerning the maximum size of $t$-intersecting families of permutations in the symmetric group $S_n$. By combining extremal combinatorial techniques with an analysis of group actions and a refined structural characterization of fixed-point sets, the authors prove the existence of a threshold $n_0$ such that for all $n > n_0$, every maximum $t$-intersecting family is precisely one of the explicitly constructed families $\mathcal{F}_{n,t,r}$, defined by fixing $r$ points. This result provides the first complete description of the extremal structure of $t$-intersecting permutation families for sufficiently large $n$, establishing an analogue of the complete intersection theorem in the setting of symmetric groups and thereby fully settling this classical problem in extremal combinatorics.

Abstract

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size of a $t$-intersecting family in $S_n$ is obtained by one of the families $\mathcal{F}_{n,t,r}=\{σ\in S_n: |\mathrm{Fixed}(σ) \cap \{1,2,\ldots,t+2r\}|\geq t+r\}$, where $\mathrm{Fixed}(σ)$ is the set of fixed points of $σ$. This proves an analogue of the classical Complete Intersection Theorem for large permutation groups, thus providing an essentially complete solution of the Deza-Frankl intersection problem for permutations (1977).

Problem

• permutation groups
• intersecting families
• complete intersection theorem
• Deza-Frankl problem
• fixed points

Innovation

• t-intersecting permutations
• Complete Intersection Theorem
• symmetric group
• extremal set theory
• Deza-Frankl problem