Multiply intersecting families of sets

Let [n] denote the set {1, 2, ..., n}, 2([n]) the collection of all subsets of [n] and F subset of 2([n]) be a family. The maximum of \F\ is studied if any r subsets have an at least s-element intersection and there are no l subsets containing t + 1 common elements. We show that \F\ less than or equal to Sigma(i=0)(t-s) ((n-s)(i)) + (t+l-s)/(t+2-s) ((n-s)(t+1-s)) + l - 2 and this bound is asymptotically the best possible as n --> infinity and t greater than or equal to 2s greater than or equal to 2, r, lgreater than or equal to2 are fixed. (C) 2004 Published by Elsevier Inc.