Frankl's diversity theorem for permutations

2026-02-12

AI Summary

This study investigates the extremal structure of non-trivial intersecting families in the symmetric group, aiming to extend Frankl’s stability theorem for set families to the setting of permutations. By combining extremal combinatorial methods, algebraic combinatorics, and representation theory of the symmetric group, the work achieves the first successful generalization of Frankl’s diversity theorem to permutations and strengthens Ellis’s Hilton–Milner-type result. For sufficiently large $n$, the authors establish a permutation analogue of Frankl’s stability theorem, precisely determining the maximum size of a non-trivial intersecting family of permutations at a fixed distance from a trivial star. This provides new structural insights and a refined perspective on extremal problems in permutation combinatorics.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

In 1987, Frankl proved an influential stability result for the Erd\H os--Ko--Rado theorem, which bounds the size of an intersecting family in terms of its distance from the nearest (subset of) star or trivial intersecting family. It is a far-reaching extension of the Hilton--Milner theorem. In this paper, we prove its analogue for permutations on $\{1,\ldots, n\}$, provided $n$ is large. This provides a similar extension of a Hilton--Milner type result for permutations proved by Ellis.

Problem

• Frankl's diversity theorem
• Erdős–Ko–Rado theorem
• intersecting families
• permutations
• Hilton–Milner theorem

Innovation

• Frankl's diversity theorem
• intersecting families
• permutations
• stability
• Erdős–Ko–Rado theorem