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Frankl's diversity theorem for permutations

Eduard Inozemtsev, Andrey Kupavskii

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.

Frankl's diversity theorem for permutations

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 , provided is large. This provides a similar extension of a Hilton--Milner type result for permutations proved by Ellis.
Paper Structure (6 sections, 9 theorems, 68 equations)

This paper contains 6 sections, 9 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The family $\mathcal{E}_k$
  3. Proof of Theorem \ref{['diversity']}
  4. Case 1. $t \geq 4k$
  5. Case 2. $t\leq 4k, k \leq \frac{\sqrt{n}}{100}$
  6. Case 3. $k\ge \sqrt n/100$, $t \leq 4k$

Key Result

Theorem 1

Let $n>2m>0$ be integers. Take an intersecting family $\mathcal{F}\subset {[n]\choose m}$ with $\gamma(\mathcal{F})\geq{n-k-1 \choose m-k}$ for some real $3\leq k\leq m$. Then and equality is possible only if $\mathcal{F}$ is isomorphic to $\mathcal{A}_k.$

Theorems & Definitions (16)

  • Theorem 1: Frankl, Frankl87; Kupavskii and Zakharov, KupZak18
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 6 more