Let
t(n, d) be the minimum number
t such that there are
t of the
n
d
lattice points
{ (x1 ,...,xd ):1 \leqslant xi \leqslant n} \{ (x_1 ,...,x_d ):1 \leqslant x_i \leqslant n\}
|
so that the (
2
t
) lines that they determine cover all the aboven
d
lattice points. We prove that for every integerd
c1 nd(d - 1)/(2d - 1) \leqslant t(n,d) \leqslant c2 nd(d - 1)/(2d - 1) logn.c_1 n^{d(d - 1)/(2d - 1)} \leqslant t(n,d) \leqslant c_2 n^{d(d - 1)/(2d - 1)} \log n.
The special cased=2 settles a problem of Erdös and Purdy.