Ting-Wei Chao (趙庭偉), Tight Bound on Joints Problem and Partial Shadow Problem

Given a set of lines in ${\mathbb{R}}^d$, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most $O(N^{3/2})$ joints in ${\mathbb{R}}^3$ via the polynomial method.

Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollobás and Eccles on the partial shadow problem. It is surprising to us that the only known proof of this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.