On families of sets without k pairwise disjoint members

For integers n≥k≥2 let m(n,k) denote the maximum size of a family of subsets of an n-set without k pairwise disjoint members. In an important classical paper Kleitman determined m(n,k) for n≡0 or −1 (mod k). We present a simple short proof of this result and prove the uniqueness of the optimal families as well.