The Existence of Designs, After Peter Keevash

This lecture explores one of the oldest and most central problems in combinatorics: the existence of designs and Steiner systems. Learn about the historical development of this mathematical challenge, from its origins with Plücker (1835), Kirkman (1846), and Steiner (1853) through to Peter Keevash's groundbreaking 2014 proof. Gil Kalai explains the fundamental question of finding collections of q-subsets from an n-element set where every r-subset appears in precisely t sets, with special focus on Steiner systems (where t=1). Discover the progression of mathematical approaches to this problem, including Richard Wilson's work for r=2, Rödl's approximate solutions, Teirlink's constructions, and finally Keevash's revolutionary "Randomized Algebraic Constructions" method that proved the existence of Steiner systems for all but finitely many admissible values of n for every q and r.