Families with no $s$ pairwise disjoint sets

For integers $n\ge s\ge 2$ let $e(n,s)$ denote the maximum of $\mathrm{\mid }\mathcal{F}\mathrm{\mid },$ where $\mathcal{F}$ is a family of subsets of an $n$-element set and $\mathcal F$containsno$s$ pairwise disjoint members. Half a century ago, solving a conjecture of Erd\H os, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for all $m,s\ge 1$. 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 $s>s_0(l,m)\mathrm{.}$ For $l=2$ we determine the value of $e(sm-2,m)$ for all $s\ge 5.$ Some related results shedding light on the problem from a more general context are proved as well.