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

TL;DR

This paper provides a concise, self-contained proof of the existence of $K_q^r$-absorbers, simplifying the original complex proof and contributing to the proof of the Existence Conjecture for Combinatorial Designs.

Contribution

It offers a short, self-contained proof of $K_q^r$-absorbers, removing reliance on previous complex constructions and advancing the proof of the Existence Conjecture.

Findings

Simplified proof of $K_q^r$-absorbers existence

Elimination of dependence on Glock, K"uhn, Lo, and Osthus's construction

Progress towards the Existence Conjecture for Combinatorial Designs

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, K\"uhn, Lo, and Osthus.