The Maximum Number of Lines Contained in Subsets of \(PG(k, 2)\)

Let \(M\) be an \(m\)-subset of \(\mathrm{PG}(k, 2)\), the finite projective geometry of dimension \(k\) over \(\mathrm{GF}(2)\). We would like to know the maximum number of lines that can be contained in \(M\). In this paper, we will not only give the maximum number of lines contained in \(m\)-subsets of \(\mathrm{PG}(k,2)\), but also construct an \(m\)-subset of \(\mathrm{PG}(k,2)\) containing the maximum number of lines.