Intersection and Diametric Problems

In this chapter we first introduce the problem of finding the maximal cardinality of a system (or family) of subsets (in particular from $\left( {\mathop k\limits^{\left[ n \right]} } \right)$), such that any two subsets from the system intersect in not less than t elements. We call such a system of subsets t-intersecting family. We also consider the diametric problem in two different spaces. The diametric problem in one of the spaces is closely connected with the intersection problem. This connection is based on a technique that was invented by Ahlswede and Khachatrian. One can understand how it works by following the proof of the Complete Intersection Theorem, which we introduce later. This technique the reader first meets in the proof of Lemma 5. It is quite different from induction or other methods known before. In some sense it is a combination of shifting, with proving of necessity of the symmetry of the family under permutations of a sufficiently large number of components. The whole method becomes clear when the reader goes through the proofs and finds out that this method allows to solve several problems that had been considered hopeless for solving before. The reward for the efforts of the reader going along the lines of rather long proofs will be the satisfaction he attains at the end.