Density results for uniform families
Assume that a \(k\)-partite family is given on some sets with \(n_ 1, \dots, n_ k\) elements. If the family has more than \(n_ 1n_ 2 \dots n_ k-(n_ 1-1) (n_ 2-1) \dots (n_ k-1)\) elements then there is a \(k\)-element set \(A\) all subsets of which occur as the intersection of some set in the family with \(A\). The proof goes by evaluating the rank of a certain matrix.
scientific article; zbMATH DE number 567476