Refined Absorption: A New Proof of the Existence Conjecture and its Applications to 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.

20 pages, to appear in Proceedings of the ICM 2026