Tight Bound and Structural Theorem for Joints

TL;DR

This paper proves the optimality and uniqueness of the known maximum number of joints formed by lines in a space, using advanced polynomial methods, and connects the problem to set-theoretic conjectures.

Contribution

It verifies the conjecture that the known construction is the only optimal one for the joints problem, establishing a tight bound and structural theorem via a sophisticated polynomial approach.

Findings

Confirmed the uniqueness of the optimal joints construction.

Established a tight bound for the maximum number of joints.

Connected the joints problem to set-theoretic conjectures and proved related conjectures.

Abstract

A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal. In this paper, we verify the conjecture and show that this is the only optimal construction by using a more sophisticated polynomial method argument. This is the first tight bound and structural theorem obtained using this method. We also give a new definition of multiplicity that strengthens the main result of a previous work by Tidor, Zhao and the second author. Lastly, we relate the joints problem to some set-theoretic problems and prove conjectures of Bollob\'{a}s and Eccles regarding partial shadows.