A family of sets is intersecting when every two of its members share at least one element. The Erdos-Ko-Rado theorem states that for n at least 2k, an intersecting family of k-element subsets of an n-set has size at most (n-1 choose k-1), attained by the star of all k-sets containing a fixed element, and uniquely so when n exceeds 2k. This concept covers the intersecting-family question in its general form: t-intersecting families and the Frankl complete-intersection theorem, cross-intersecting pairs, stability and Hilton-Milner-type results bounding non-star families, and the transfer of the question from set systems to permutations, vector spaces over finite fields, and the integers. It also covers the four proof techniques that recur across these settings - the shifting or compression argument, the cycle method, the Hoffman ratio bound applied to a Kneser-type association scheme, and Fourier or spectral analysis on the discrete cube - and what each one buys in terms of sharpness, stability, and range of validity.
A survey of intersection problems in extremal combinatorics: how large a family of finite objects can be if any two members of the family overlap in a prescribed way. The central object is an interse…