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Erd\H os--Ko--Rado type results for partitions via spread approximations

Andrey Kupavskii

Abstract

In this paper, we address several Erd\H os--Ko--Rado type questions for families of partitions. Two partitions of $[n]$ are {\it $t$-intersecting} if they share at least $t$ parts, and are {\it partially $t$-intersecting} if some of their parts intersect in at least $t$ elements. The question of what is the largest family of pairwise $t$-intersecting partitions was studied for several classes of partitions: Peter Erd\H os and Székely studied partitions of $[n]$ into $\ell$ parts of unrestricted size; Ku and Renshaw studied unrestricted partitions of $[n]$; Meagher and Moura, and then Godsil and Meagher studied partitions into $\ell$ parts of equal size. We improve and generalize the results proved by these authors. Meagher and Moura, following the work of Erd\H os and Székely, introduced the notion of partially $t$-intersecting partitions, and conjectured, what should be the largest partially $t$-intersecting family of partitions into $\ell$ parts of equal size $k$. The main result of this paper is the proof of their conjecture for all $t, k$, provided $\ell$ is sufficiently large. All our results are applications of the spread approximation technique, introduced by Zakharov and the author. In order to use it, we need to refine some of the theorems from the original paper. As a byproduct, this makes the present paper a self-contained presentation of the spread approximation technique for $t$-intersecting problems.

Erd\H os--Ko--Rado type results for partitions via spread approximations

Abstract

In this paper, we address several Erd\H os--Ko--Rado type questions for families of partitions. Two partitions of are {\it -intersecting} if they share at least parts, and are {\it partially -intersecting} if some of their parts intersect in at least elements. The question of what is the largest family of pairwise -intersecting partitions was studied for several classes of partitions: Peter Erd\H os and Székely studied partitions of into parts of unrestricted size; Ku and Renshaw studied unrestricted partitions of ; Meagher and Moura, and then Godsil and Meagher studied partitions into parts of equal size. We improve and generalize the results proved by these authors. Meagher and Moura, following the work of Erd\H os and Székely, introduced the notion of partially -intersecting partitions, and conjectured, what should be the largest partially -intersecting family of partitions into parts of equal size . The main result of this paper is the proof of their conjecture for all , provided is sufficiently large. All our results are applications of the spread approximation technique, introduced by Zakharov and the author. In order to use it, we need to refine some of the theorems from the original paper. As a byproduct, this makes the present paper a self-contained presentation of the spread approximation technique for -intersecting problems.
Paper Structure (14 sections, 18 theorems, 46 equations)

This paper contains 14 sections, 18 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. General partitions
  3. Partitions with $k$ parts
  4. Partitions with fixed profile
  5. Partially $t$-intersecting partitions
  6. The structure of the remainder of the paper
  7. Sketches of the proofs
  8. Spread approximations
  9. Proofs
  10. General partitions. Proof of Theorem \ref{['thm6']}
  11. Partitions with $k$ parts. Proof of Theorem \ref{['thmp1']}
  12. Profiled $t$-intersecting partitions. Proof of Theorem \ref{['thmu1']}
  13. Partially $t$-intersecting partitions. Proof of Theorem \ref{['thm1']}
  14. Acknowledgements

Key Result

Theorem 1

Let $n\ge n_0(t)$ and assume that $\mathcal{F}\subset \mathcal{B}_n$ is $t$-intersecting. Then $|\mathcal{F}|\le B_{n-t}$, with equality only possible if $\mathcal{F}$ is a family of all partitions with $t$ fixed singletons.

Theorems & Definitions (31)

  • Theorem 1: Ku and Renshaw KuRe
  • Theorem 2
  • Theorem 3: Erdős and Székely ErSz
  • Theorem 4
  • Theorem 5: MeMo
  • Theorem 6: MeMo
  • Theorem 7
  • Conjecture 1: Meagher and Moura, MeMo
  • Theorem 8
  • Theorem 9
  • ...and 21 more