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Refined Absorption: A New Proof of the Existence Conjecture

Michelle Delcourt, Luke Postle

Abstract

The study of combinatorial designs has a rich history spanning nearly two centuries. In a recent breakthrough, the notorious Existence Conjecture for Combinatorial Designs dating back to the 1800s was proved in full by Keevash via the method of randomized algebraic constructions. Subsequently Glock, Kühn, Lo, and Osthus provided an alternate purely combinatorial proof of the Existence Conjecture via the method of iterative absorption. We introduce a novel method of refined absorption for designs; here as our first application of the method we provide a new alternate proof of the Existence Conjecture (assuming the existence of $K_q^r$-absorbers by Glock, Kühn, Lo, and Osthus).

Refined Absorption: A New Proof of the Existence Conjecture

Abstract

The study of combinatorial designs has a rich history spanning nearly two centuries. In a recent breakthrough, the notorious Existence Conjecture for Combinatorial Designs dating back to the 1800s was proved in full by Keevash via the method of randomized algebraic constructions. Subsequently Glock, Kühn, Lo, and Osthus provided an alternate purely combinatorial proof of the Existence Conjecture via the method of iterative absorption. We introduce a novel method of refined absorption for designs; here as our first application of the method we provide a new alternate proof of the Existence Conjecture (assuming the existence of -absorbers by Glock, Kühn, Lo, and Osthus).
Paper Structure (30 sections, 25 theorems, 103 equations)

This paper contains 30 sections, 25 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The Existence Conjecture for Combinatorial Designs
  3. Refined Absorbers
  4. Refiners
  5. Single-Step Absorption (and Edge Vortex) vs Iterative Absorption
  6. Outline of Paper
  7. Notation and Standard Definitions
  8. Proof Overview
  9. Building Refiners
  10. Nibble with Reserves
  11. Multi-Refiners
  12. Multi-Refiners with Remainder
  13. Proof Overview of Multi-Refiner Theorem
  14. Multiplicity Reduction
  15. Proof of Edge Sparsification Lemma
  16. ...and 15 more sections

Key Result

Theorem 1.2

Conjecture conj:Existence is true.

Theorems & Definitions (73)

  • Conjecture 1.1: Existence Conjecture
  • Theorem 1.2: Keevash K14
  • Definition 1.3: Absorber
  • Theorem 1.4: Glock, Kühn, Lo, and Osthus GKLO16
  • Definition 1.5: Omni-Absorber
  • Definition 1.6: Refined Omni-Absorber
  • Theorem 1.7: Refined Omni-Absorber Theorem
  • Definition 1.8: Refiner
  • Definition 1.9: $C$-Refined Family
  • Definition 1.10: $C$-Refined Refiner
  • ...and 63 more