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On the intersecting family process

Patrick Bennett, Alan Frieze, Andrew Newman, Wesley Pegden

Abstract

We study the intersecting family process initially studied in \cite{BCFMR}. Here $k=k(n)$ and $E_1,E_2,\ldots,E_m$ is a random sequence of $k$-sets from $\binom{[n]}{k}$ where $E_{r+1}$ is uniformly chosen from those $k$-sets that are not already chosen and that meet $E_i,i=1,2,\ldots,r$. We prove some new results for the case where $k=cn^{1/3}$ and for the case where $k\gg n^{1/2}$.

On the intersecting family process

Abstract

We study the intersecting family process initially studied in \cite{BCFMR}. Here and is a random sequence of -sets from where is uniformly chosen from those -sets that are not already chosen and that meet . We prove some new results for the case where and for the case where .
Paper Structure (18 sections, 12 theorems, 109 equations, 1 algorithm)

This paper contains 18 sections, 12 theorems, 109 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm2']}
  3. The first $o(\sqrt{\log n})$ steps: generating $\mathcal{S}$
  4. Step $o(\sqrt{\log n})$ to step $n^{\omega(1)}$
  5. From step $n^{\omega(1)}$ onward
  6. The size of $\mathcal{I}^*$ and $\mathcal{I}^{**}$
  7. Proof of Theorem \ref{['th2']} when $k =\Theta(n)$
  8. The good event
  9. Dynamic concentration of $V(r)$
  10. Dynamic concentration of $D_v(r)$
  11. The upper bound on $C_v$
  12. Proof of Theorem \ref{['th2']} when $k=o(n)$
  13. The good event
  14. Dynamic concentration of $V(r)$
  15. Dynamic concentration of $D_v(r)$
  16. ...and 3 more sections

Key Result

Theorem 1

Let ${\mathcal{E}}_0$ be the event that ${\cal I}_k={\mathcal{A}}_x$ for some $x\in [n]$. If $k=c_nn^{1/3}<n/2$ then

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • proof
  • Definition 1
  • Definition 2
  • Lemma 7
  • ...and 11 more