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An almost complete $t$-intersection theorem for permutations

Andrey Kupavskii

TL;DR

This work determines the maximal size of a $t$-intersecting family of permutations in $\\Sigma_n$ for the regime $n>(1+\eta)t$ and proves a sharp stability result. It introduces and refines spread-approximation techniques, coupled with a peeling procedure, to reveal that large families are structurally governed by collections $\\mathcal{A}_k$, yielding an Ahlswede–Khachatrian–type classification for permutations. The results resolve the Ellis–Friedgut–Pilpel conjecture in this range and confirm conjectures of Frankl–Deza and Cameron, with the latter showing that extremal examples can be nontrivial. The methods provide a robust, near-complete description of forbidden-intersection phenomena for permutations and may extend to other combinatorial structures, highlighting the broader impact of spread-based and density-augmentation approaches.

Abstract

For any $ε>0$ and $n>(1+ε)t$, $n>n_0(ε)$ we determine the size of the largest $t$-intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) for $n>(1+ε)t$. We note that, for this range of parameters, the extremal examples are not necessarily trivial, and that our statement is analogous to the celebrated Ahlswede-Khachatrian theorem. The proof is based on the refinement of the method of spread approximations, recently introduced by Kupavskii and Zakharov (2022).

An almost complete $t$-intersection theorem for permutations

TL;DR

This work determines the maximal size of a -intersecting family of permutations in for the regime and proves a sharp stability result. It introduces and refines spread-approximation techniques, coupled with a peeling procedure, to reveal that large families are structurally governed by collections , yielding an Ahlswede–Khachatrian–type classification for permutations. The results resolve the Ellis–Friedgut–Pilpel conjecture in this range and confirm conjectures of Frankl–Deza and Cameron, with the latter showing that extremal examples can be nontrivial. The methods provide a robust, near-complete description of forbidden-intersection phenomena for permutations and may extend to other combinatorial structures, highlighting the broader impact of spread-based and density-augmentation approaches.

Abstract

For any and , we determine the size of the largest -intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) for . We note that, for this range of parameters, the extremal examples are not necessarily trivial, and that our statement is analogous to the celebrated Ahlswede-Khachatrian theorem. The proof is based on the refinement of the method of spread approximations, recently introduced by Kupavskii and Zakharov (2022).
Paper Structure (8 sections, 10 theorems, 36 equations)

This paper contains 8 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thmbigt']}
  3. Sketch of the proof.
  4. Preliminaries
  5. First spread approximation
  6. The peeling procedure and the structure of spread approximations for large $t$-intersecting families
  7. Finding a dense piece of $\mathcal{F}$ and a sharper spread approximation
  8. Proof of Theorem \ref{['thmbigt']}

Key Result

Theorem 1

Let $\eta>0$ be some constant and $n,t$ be integers such that $n>(1+\eta)t$ and $n\geqslant n_0(\eta)$. If $\mathcal{F}\subset \Sigma_n$ is $t$-intersecting then $|\mathcal{F}| \leqslant \max_{k\in \mathbb{N}} |\mathcal{A}_k|$. Moreover, if $\mathcal{F}$ is not fully contained in some $\mathcal{F}\i

Theorems & Definitions (20)

  • Conjecture 1
  • Theorem 1
  • proof
  • Theorem 4: Alw, a sharpening due to Tao
  • Lemma 5
  • proof
  • Theorem 6
  • proof : Proof of Theorem \ref{['thm2']}
  • Lemma 7
  • proof
  • ...and 10 more