Economical coverings of sets of lattice points (Q1190135)

Let \(t(n,d)\) be the minimal number \(t\) such that there are \(t\) of the \(n^ d\) lattice points \(\{(x_ 1,\dots,x_ d): 1\leq i\leq n\}\) so that the \(t\choose 2\) lines defined by them cover all these \(n^ d\) lattice points. The author proves for \(d\geq 2\) and some \(c_ 1=c_ 1(d)\), \(c_ 2=c_ 2(d)\): \[ c_ 1n^{d(d-1)(2d-1)}\leq t(n,d)\leq c_ 2n^{d(d- 1)(2d-1)}\log n \] for each \(n\).