Extremal set systems with weakly restricted intersections
In this excellent paper extremal set systems with intersections of restricted cardinality are studied. If, say, each intersection has even cardinality then a linear algebraic tool (linear independence of the characteristic vectors of the subsets) gives an upper bound for the number of subsets. If, for each \(i\), the cardinality constraints for \(A_i \cap A_j\) can be violated by at most \(s\) subsets of form \(A_j\) then this ``\(s\)-weak version'' of the problem can still be handled by additional tools (from extremal graph theory, for instance) if \(s\) is relatively small. The author gives a tight upper bound for the weak version of the ``odd town'' problem (odd subsets with even intersections); for the weak version of the non-uniform Fisher inequality (leading to a new, extremal set theoretic characterization of Hadamard's matrices); and for the weak version of the ``even town'' problem (even intersections only). This latter leads to results in case of restricted multi-intersections as well.
scientific article; zbMATH DE number 1446813