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Loose paths in random ordered hypergraphs

Andrzej Dudek, Alan Frieze, Wesley Pegden

TL;DR

This work investigates the maximum length $\ell_{\max}$ of ordered loose paths in the random $r$-uniform hypergraph $H^{(r)}(n,p)$. It combines greedy path-extension, first- and second-moment methods, McDiarmid concentration, and Warnke’s inequality to derive tight high-probability bounds across diverse $p$-regimes, and it proves Poisson limits for the count of fixed-length paths. The main findings show that, in wide ranges of $p$, $\ell_{\max}=\Theta\big(n p^{1/(r-1)}\big)$, with precise thresholds and refinements in intermediates regimes, including a Poisson convergence phenomenon when $p$ is of order $n^{-(r-1+1/\ell)}$. These results extend the graph-path-length paradigm to hypergraphs, clarifying phase transitions in path length as a function of edge probability and offering a nuanced map of regime-dependent behavior.

Abstract

We consider the length of {\em ordered loose paths} in the random $r$-uniform hypergraph $H=H^{(r)}(n, p)$. A ordered loose path is a sequence of edges $E_1,E_2,\ldots,E_\ell$ where $\max\{j\in E_i\}=\min\{j\in E_{i+1}\}$ for $1\leq i<\ell$. We establish fairly tight bounds on the length of the longest ordered loose path in $H$ that hold with high probability.

Loose paths in random ordered hypergraphs

TL;DR

This work investigates the maximum length of ordered loose paths in the random -uniform hypergraph . It combines greedy path-extension, first- and second-moment methods, McDiarmid concentration, and Warnke’s inequality to derive tight high-probability bounds across diverse -regimes, and it proves Poisson limits for the count of fixed-length paths. The main findings show that, in wide ranges of , , with precise thresholds and refinements in intermediates regimes, including a Poisson convergence phenomenon when is of order . These results extend the graph-path-length paradigm to hypergraphs, clarifying phase transitions in path length as a function of edge probability and offering a nuanced map of regime-dependent behavior.

Abstract

We consider the length of {\em ordered loose paths} in the random -uniform hypergraph . A ordered loose path is a sequence of edges where for . We establish fairly tight bounds on the length of the longest ordered loose path in that hold with high probability.

Paper Structure

This paper contains 16 sections, 10 theorems, 85 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower bound for $\sqrt{\log n}/n^{(r-1)/4}\ll p \le 1-o(1)$.
  4. Lower bound for $\Omega(1/n^{(r-1)/2}) = p = O(\sqrt{\log n}/n^{(r-1)/4})$.
  5. Lower bound for $\Omega((\log n)^{r-1}/n^{r-1}) = p \ll 1/n^{(r-1)/2})$.
  6. Upper bound for $\Omega((\log n)^{r-1}/n^{r-1}) = p$.
  7. Bounding the union of paths
  8. First moment method
  9. Lower and upper bound for $\frac{1}{n^{r-1+1/\omega}} \le p \le O((\log n)^{r-1}/n^{r-1})$.
  10. Upper bound
  11. Lower bound
  12. Lower and upper bound for $1/(n^{r-1+1/\ell}) \ll p \ll 1/(n^{r-1+1/(\ell+1)})$ where $\ell=O(1)$.
  13. Upper bound
  14. Lower bound
  15. Method of moments
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $r\ge 2$ and $\Omega(1)=p\le 1-o(1)$. Then, a.a.s.

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Remark 1
  • ...and 9 more