More about shifting techniques

We discovered a new and simple shifting technique. It makes it possible to prove results on shadows like the Kruskal–Katona theorem without any additional arguments. ther application we obtain the following new result. For s, d, k∈N, 1≤d≤s, d≤k define the subclass of Nk (the k-subsets of N) B(k,s,d)=B∈Nk:|B∩[1,s]|≥d. Let A⊂B(k,s,d) and |A|=m. Then the cardinality of the ℓ-shadow of A is minimal if A consists of the first m elements of B(k,s,d) in colexicographic order. A more general form of this result is given as well. Other applications are to be expected.