Extremal Set Systems with Weakly Restricted Intersections

Theorems on extremal set systems with intersections of restricted cardinality are probably among the most well-known and powerful theorems in extremal combinatorics. In this paper, we discuss the so-call ”weak” version of some theorems of this type, when the restricted intersection property is weakened by the possible existence of some (maybe many) intersections having ”bad” sizes. In particular, we give a tight upper bound for the weak version of the ”odd town” problem and non-uniform Fisher’s inequality. The second problem leads to an extremal set theoretic characterization of Hadamard’s matrices. We also give a tight bound for the weak version of the ”even town” problem, raised by Erdős. This bound turns out to be an useful tool to handle systems with restricted multi-intersections.