Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory
Luke Postle
TL;DR
The paper provides a consolidated, combinatorial framework—refined absorption—for proving the Existence Conjecture in combinatorial design theory and for tackling central extremal and probabilistic design questions. It introduces a refined, efficient omni-absorber as a black-box tool, together with a nibble-with-reserves mechanism and a powerful embedding toolkit, to convert almost-structures into exact decompositions. The authors apply this framework to reprove the Existence Conjecture and to advance results on Nash-Williams’ conjecture, high-girth Steiner systems, and threshold phenomena for Steiner systems in random settings, unifying several strands of design theory under a single method. The work paves the way for a unified theorem of designs and offers a modular approach to embedding and refinement that can adapt to multiple structural constraints (girth, spread, partite settings) with broad implications for extremal and probabilistic design theory.
Abstract
We discuss the recently developed method of refined absorption and how it is used to provide a new proof of the Existence Conjecture for combinatorial designs. This method can also be applied to resolve open problems in extremal and probabilistic design theory while providing a unified framework for these problems. Crucially, the main absorption theorem can be used as a "black-box" in these applications obviating the need to reprove the absorption step for each different setup.