Combinatorial Properties of Systems of Sets

A family of sets {A k } is called a strong d system if the intersection of any two of its members is the same, i .e ., if A kI n a = A ka n A ka . It is called a weak d system if I A k= rn A k . 1 is the same for any two sets of our family . J systems have recently been studied in several papers . f (n, r) is the smallest integer for which any family of f (n, r) sets A k , I < k < f (n, r) of size n I Ak I = n, 1 < k < f (n, r) contains a subfamily of r sets {A kt } 1 < I < r which form a strong d system, g(n, r) is the smallest integer for which every family of g(n, r) sets Ak , 1 ~ k g(n, r) of size n contains a subfamily of r sets {A kt }, 1 I-~; r which form a weak d system . Erdős and Rado [1] proved