Families with no s pairwise disjoint sets

For integers n >= s >= 2 let e (n, s) denote the maximum of vertical bar F vertical bar, where F is a family of subsets of an n- element set and F contains no s pairwise disjoint members. Half a century ago, solving a conjecture of Erd. os, Kleitman determined e(sm - 1, s) and e (sm, s) for all m, s >= 1. During the years very little progress in the general case was made. In the present paper we state a general conjecture concerning the value of e(sm - l, m) for 1 < l < s and prove its validity for s > s(0) (l, m). For l = 2 we determine the value of e(sm - 2, m) for all s >= 5. Some related results shedding light on the problem from a more general context are also proved.