The distinct-direction conjecture for point sets in Euclidean space

Let a $d$-dimensional set of $n$ points in $obreak\mathbb{R}^d$ be a set whose affine span has dimension $d$. Two lines have pairwise distinct directions when no two are parallel. The distinct-direction conjecture. A $d$-dimensional set of $n$ points determines at least

$$(d-1)n-O(d^2)$$

lines with pairwise distinct directions. This conjecture is suggested by work of Jamison, Blokhuis and Seress, and by the known sharp results in dimensions two and three; it remains open, although several weaker versions have been established.