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Proof of the High Girth Existence Conjecture via Refined Absorption

Michelle Delcourt, Luke Postle

Abstract

We prove the High Girth Existence Conjecture - the common generalization of the Existence Conjecture for Combinatorial Designs originating from the 1800s and Erdős' Conjecture from 1973 on the Existence of High Girth Steiner Triple Systems.

Proof of the High Girth Existence Conjecture via Refined Absorption

Abstract

We prove the High Girth Existence Conjecture - the common generalization of the Existence Conjecture for Combinatorial Designs originating from the 1800s and Erdős' Conjecture from 1973 on the Existence of High Girth Steiner Triple Systems.
Paper Structure (38 sections, 38 theorems, 250 equations)

This paper contains 38 sections, 38 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Existence Conjecture
  3. High Girth Steiner Systems
  4. Refined Absorbers
  5. High Level Proof Overview
  6. Outline of Paper
  7. Notation
  8. Proof of the High Girth Existence Conjecture
  9. Random Subsets and Regularity Boosting
  10. Forbidden Submatching with Reserves
  11. Configuration Hypergraphs for High Girth Designs
  12. Girth-$g$ Projections for Omni-Absorbers
  13. Proof of the High Girth Existence Conjecture
  14. High Girth Boosters
  15. The Necessity of High Cogirth Boosters
  16. ...and 23 more sections

Key Result

Theorem 1.2

Conjecture conj:Existence is true.

Theorems & Definitions (130)

  • Conjecture 1.1: Existence Conjecture
  • Theorem 1.2: Keevash K14
  • Conjecture 1.3: Existence of High Girth Steiner Triple Systems - Erdős E73
  • Theorem 1.4: Kwan, Sah, Sawhney, and Simkin KSSS22
  • Conjecture 1.5: Existence of High Girth Designs - Glock, Kühn, Lo, and Osthus GKLO20; Keevash and Long KL20
  • Theorem 1.6: Existence of High Girth Designs
  • Remark 1.7
  • Definition 1.8: Absorber
  • Definition 1.9: Omni-Absorber
  • Definition 1.10: Refined Omni-Absorber
  • ...and 120 more