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The Complete Intersection Theorem

The complete intersection theorem gives the answer to the question: What is the maximum size of a family of k-subsets of [n] = {1, . . . , n} such that every pair of sets from the family has intersection size at least t? This is an extension of the Erdős-Ko-Rado Theorem, which says that in the case of t = 1 and n ≥ 2k, the answer is (n−1 k−1). This theorem and the next show that upto permutations, the maximum family in all cases (except if n = 2k, t = 1) has the property that the required t-intersection occurs in the first t+ 2r elements, for a particular r defined later. This type of family of sets is denoted