Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Thresholds for $(n,q,2)$-Steiner Systems via Refined Absorption

Michelle Delcourt, Tom Kelly, Luke Postle

Abstract

We prove that if $p \geq n^{-(q-6)/2}$, then asymptotically almost surely the binomial random $q$-uniform hypergraph $G^{(q)}(n,p)$ contains an $(n,q,2)$-Steiner system, provided $n$ satisfies the necessary divisibility conditions.

Thresholds for $(n,q,2)$-Steiner Systems via Refined Absorption

Abstract

We prove that if , then asymptotically almost surely the binomial random -uniform hypergraph contains an -Steiner system, provided satisfies the necessary divisibility conditions.
Paper Structure (17 sections, 12 theorems, 36 equations)

This paper contains 17 sections, 12 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Proof Overview
  3. Proof of Existence via Refined Absorption
  4. Building a Spread Omni-Absorber
  5. Outline of Paper
  6. Preliminaries
  7. Reservoir
  8. The Design Hypergraph
  9. Nibble with Reserves
  10. The Lovász Local Lemma Distribution
  11. Spread Omni-Absorber Theorem
  12. Spread Boosters
  13. Vertex-Spread Clique Embedding
  14. Proof of the Spread Omni-Absorber Theorem
  15. Construction of Spread Boosters
  16. ...and 2 more sections

Key Result

Theorem 1.3

For every integer $q > 2$, the following holds: If $q - 1 \mid n - 1$ and $\binom{q}{2}\mid \binom{n}{2}$ and $p\geq n^{-(q-6)/2}$, then asymptotically almost surely $\mathcal{G}^{(q)}(n,p)$ contains an $(n, q, 2)$-Steiner system.

Theorems & Definitions (27)

  • Definition 1.1
  • Conjecture 1.2: Kang, Kelly, Kühn, Methuku, and Osthus KKKMO22
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1: Omni-Absorber
  • Definition 2.2: Refined Omni-Absorber
  • Theorem 2.3
  • Lemma 3.1
  • Definition 3.2: Design Hypergraph
  • ...and 17 more