On December 12, 2023, Ting-Wei Chao (趙庭偉) from Carnegie Mellon University gave a talk at the Discrete Math Seminar on the number of points that are intersections of d linearly independent lines among given n lines in the d-dimensional space. The title of his talk was “Tight Bound on Joints Problem and Partial Shadow Problem“.

## 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.