Exact k-Wise Intersection Theorems

A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k − 1 is a power of 2, the size of the extremal k-wise oddtown family is (k − 1)(n− 2 log 2 (k − 1)). Tight bounds are also found in several other basic cases.