Intersection Theorems for Finite Sets
Finite extremal set theory is concerned with the following general problem: Suppose we have a collection F of subsets of an n-element set and we have some restriction on the possible intersection sizes of pairs of sets in F. What is the maximum number of subsets that F can contain? Surprisingly, solutions to various special cases of this problem have deep implications in many other areas, including coding theory, geometry, and computer science. A particular famous example is due to Frankl and Rodl, who solved a 250-dollar problem of Erdos by proving that if n is a multiple of 4 and n/4 is excluded as an intersection size, then |F| < (1.99)^n. We extend this result by showing that if some additional (rather mild) restrictions are placed on the possible intersection sizes, then |F|< (1.63)^n. This is joint work with Vojtech Rodl.