Intersecting balanced families of sets

Suppose that any t members (t greater than or equal to 2) of a regular family on and element set have ill least k common elements. It is proved that the largest member of the family has at least k(1/2)n(1-1/t) elements. The same holds for balanced families. which is a generalization of the regularity. The estimate is asymptotically sharp. (C) 2001 Academic Press.