FAMILIES OF FINITE SETS WITH MISSING INTERSECTIONS

Abstract

Let X be an n-element set, l, s positive integers satisfying s ≥ ≥ 3l+2. Suppose that F is a family of subsets of X having the property that for any two different F, F′ ∈F we have |F∪F‘|≥ s and |F ∪ F′ | ≠ l We prove that |F| ≥ (1+ 0 (1))(n-1-1s-1). Taking all the (s+1)-sets containing a given (l+1)-set shows that this is asymptotically best possible.