Refined Absorption: A New Proof of the Existence Conjecture

The study of combinatorial designs has a rich history spanning nearly two centuries. In a recent breakthrough, the notorious Existence Conjecture for Combinatorial Designs dating back to the 1800s was proved in full by Keevash via the method of randomized algebraic constructions. Subsequently Glock, Kühn, Lo, and Osthus provided an alternate purely combinatorial proof of the Existence Conjecture via the method of iterative absorption. We introduce a novel method of refined absorption for designs; here as our first application of the method we provide a new alternate proof of the Existence Conjecture (assuming the existence of K_q^r-absorbers by Glock, Kühn, Lo, and Osthus).