A Complete Intersection Theorem for Large Permutation Groups
2026-07-01 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study collections of permutations where any two permutations agree on at least t points. They show that for large enough n, the biggest such collections are formed by permutations fixing a specific number of points in a certain initial segment. This result extends a well-known intersection theorem to permutations and solves a long-standing problem proposed by Deza and Frankl in 1977. Essentially, they identify the largest families of permutations with strong overlap properties for large sets.
permutationt-intersecting familyfixed pointsComplete Intersection TheoremDeza-Frankl problemsymmetric groupintersection theoremcombinatorics
Authors
Nathan Keller, Andrey Kupavskii, Noam Lifshitz, Ohad Sheinfeld
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).