The Joint Problem

Joint

Let $\mathcal{L}$ be a set of lines in ${\mathbb{R}}^3$. A joint of $\mathcal{L}$ is a point which lies in three non-coplanar lines of $\mathcal{L}$.

The joints problem asks what is the maximal number of joints that can be formed from L lines.

Lemma

If L is a set of lines in ${\mathbb{R}}^3$ that determines $J$ joints, then one of the lines contains at most $3J^{\frac{1}{3}}$ joints.

Theorem

Any L lines in space determine $\le 10L^{\frac{3}{2}}$ joints.