 Joints theorem

The number of joints of a line collection formed by $L$ distinct real affine lines in ${\mathbb{R}}^n$ is at most $C_n{L}^{n/(n-1)}$. The pruning and minimal-degree polynomial argument proves this bound. The exponent is sharp: the coordinate-line grid has $m^n$ joints of a line collection and $n{m}^{n-1}$ lines. In three dimensions the bound is $C{L}^{3/2}$.