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.