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A Short Proof of the Existence of $K_q^r$-absorbers

Michelle Delcourt, Tom Kelly, Luke Postle

TL;DR

This work provides a concise, self-contained proof of the existence of $K_q^r$-absorbers, a key ingredient in the Existence Conjecture for combinatorial designs. It replaces prior randomized algebraic methods with a layered construction of orthogonal boosters, leveraging a Cauchy-matrix framework to produce multiple disjoint $K_q^r$-decompositions and then assembling an absorber via an integral $K_q^r$-decomposition of the target graph. The absorber $A$ for a $K_q^r$-divisible graph $L$ is built from boosters and independent hinges in a way that yields decompositions of both $A$ and $A L$, proving the absorber's existence and functionality. The result strengthens the combinatorial toolkit for the Existence Conjecture and aligns with Keevash’s layering techniques, offering improved quantitative properties and variant applicability without relying on Glock–Kühn–Lo–Osthus-style absorber constructions.

Abstract

We codify a short self-contained proof of the existence of $K_q^r$-absorbers implicit in Keevash's original proof of the Existence Conjecture. Combining this with the work of the first and third authors in yields a proof of the Existence Conjecture for Combinatorial Designs that is not reliant on the construction of $K_q^r$-absorbers by Glock, Kühn, Lo, and Osthus.

A Short Proof of the Existence of $K_q^r$-absorbers

TL;DR

This work provides a concise, self-contained proof of the existence of -absorbers, a key ingredient in the Existence Conjecture for combinatorial designs. It replaces prior randomized algebraic methods with a layered construction of orthogonal boosters, leveraging a Cauchy-matrix framework to produce multiple disjoint -decompositions and then assembling an absorber via an integral -decomposition of the target graph. The absorber for a -divisible graph is built from boosters and independent hinges in a way that yields decompositions of both and , proving the absorber's existence and functionality. The result strengthens the combinatorial toolkit for the Existence Conjecture and aligns with Keevash’s layering techniques, offering improved quantitative properties and variant applicability without relying on Glock–Kühn–Lo–Osthus-style absorber constructions.

Abstract

We codify a short self-contained proof of the existence of -absorbers implicit in Keevash's original proof of the Existence Conjecture. Combining this with the work of the first and third authors in yields a proof of the Existence Conjecture for Combinatorial Designs that is not reliant on the construction of -absorbers by Glock, Kühn, Lo, and Osthus.

Paper Structure

This paper contains 2 sections, 5 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof

Key Result

Theorem 1.2

Let $q > r\ge 1$ be integers. If $L$ is a $K_q^r$-divisible $r$-graph, then there exists a $K_q^r$-absorber for $L$.

Theorems & Definitions (12)

  • Definition 1.1: Absorber
  • Theorem 1.2: Glock, Kühn, Lo, and Osthus GKLO16
  • Definition 2.1: Booster
  • Definition 2.2: Hinge
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 2 more