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Algebraic approach to stability results for Erdős-Ko-Rado theorem

Gennian Ge, Zixiang Xu, Xiaochen Zhao

Abstract

Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erdős, Ko, and Rado in 1961 established that any $k$-uniform intersecting family on $[n]$ has a maximum size of $\binom{n-1}{k-1}$, with the unique extremal structure being a star. In 1967, Hilton and Milner followed up with a pivotal result, showing that if such a family is not a star, its size is at most $\binom{n-1}{k-1} - \binom{n-k-1}{k-1} + 1$, and they identified the corresponding extremal structures. In recent years, Han and Kohayakawa, Kostochka and Mubayi, and Huang and Peng have provided the second and third levels of stability results in this line of research. In this paper, we provide a unified approach to proving the stability result for the Erdős-Ko-Rado theorem at any level. Our framework primarily relies on a robust linear algebra method, which leverages appropriate non-shadows to effectively handle the structural complexities of these intersecting families.

Algebraic approach to stability results for Erdős-Ko-Rado theorem

Abstract

Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erdős, Ko, and Rado in 1961 established that any -uniform intersecting family on has a maximum size of , with the unique extremal structure being a star. In 1967, Hilton and Milner followed up with a pivotal result, showing that if such a family is not a star, its size is at most , and they identified the corresponding extremal structures. In recent years, Han and Kohayakawa, Kostochka and Mubayi, and Huang and Peng have provided the second and third levels of stability results in this line of research. In this paper, we provide a unified approach to proving the stability result for the Erdős-Ko-Rado theorem at any level. Our framework primarily relies on a robust linear algebra method, which leverages appropriate non-shadows to effectively handle the structural complexities of these intersecting families.

Paper Structure

This paper contains 10 sections, 8 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Robust linear algebra methods
  4. Some useful lemmas
  5. Overview of the robust linear algebra methods
  6. A unified framework and proof of \ref{['thm:AllLayerStability']}
  7. High level overview of our framework
  8. Our framework
  9. Stability at arbitrary level: Proof of \ref{['thm:AllLayerStability']}
  10. Concluding remarks

Key Result

Theorem 1.1

Let $n,k$ be positive integers with $n\ge 2k$. If $\mathcal{F}\subseteq \binom{[n]}{k}$ is an intersecting family, then we have The equality holds if and only if $\mathcal{F}=\{F\in\binom{[n]}{k}:p\in F\}$ for some $p\in [n]$.

Theorems & Definitions (22)

  • Theorem 1.1: Erdős-Ko-Rado 1961EKR
  • Theorem 1.2: Hilton-Milner 1967Hilton
  • Theorem 1.3: Han-Kohayakawa 2017PAMSHanJie
  • Theorem 1.4: Huang-Peng 2024EUJCPeng
  • Theorem 1.5: $t$-th level stability
  • Proposition 2.1
  • Lemma 2.2
  • proof : Proof of \ref{['HowToLP']}
  • Theorem 2.3
  • Claim 3.1
  • ...and 12 more