Bounding one-way differences (Q1114687)

Let f(n,k) denote the maximum length of a sequence \((F_ 1,F_ 2,...)\) of distinct subsets of an n-element set with the property that \(| F_ i\setminus F_ j| <k\) for all \(i<j\). We determine the exact values of f(n,2) and characterize all the extremal sequences. For \(k\geq 3\) we prove that \(f(n,k)=(1+o(1))\left( \begin{matrix} n\\ k\end{matrix} \right).\) Some related problems are also considered.