Extremal Set Theory

Peter Keevash
Queen Mary, University of London

Questions concerning the maximum size of set systems with certain properties have a rich history in Combinatorics. Classical results include Sperner's theorem on the maximum size of an antichain in the Boolean lattice and the Erd\H{o}s-Ko-Rado theorem on the maximum size of an intersecting uniform family. These can be proved either by compression arguments or probabilistic methods, both of which are instructive and generalise in different ways.
Algebraic methods also play a valuable role, most notably in the Frankl-Ray-Chaudhuri-Wilson theorems on restricted intersections, which have dramatic applications in diverse areas, including geometry and topology. More recently, spectral methods have led to structural results on intersection problems. We will discuss these methods and some open problems.

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