For all n>n_a, there are at most 2n-2 lines in R^n with common angle a

You need to know: Euclidean space ${\mathbb R}^n$ , origin in ${\mathbb R}^n$ , lines in ${\mathbb R}^n$ , angle between lines, the inverse trigonometric function of cosine $\arccos$ .

Background: A set of lines through the origin in ${\mathbb R}^n$ is called equiangular if any pair of lines defines the same angle. Let $N(n)$ denotes the maximum cardinality of an equiangular set of lines in ${\mathbb R}^n$ . Let $N_\theta(n)$ denotes the maximum number of equiangular lines in ${\mathbb R}^n$ with common angle $\theta$ , where $\theta$ does not depend on dimension.

The Theorem: On 21st June 2016, Igor Balla, Felix Dräxler, Peter Keevash, and Benny Sudakov submitted to arxiv a paper in which they proved that for any angle $\theta \in (0,\pi/2)$ , $\theta \neq \arccos \frac{1}{3}$ , there is a constant $n_\theta$ , such that $N_\theta(n)\leq 1.93n$ for all $n \geq n_\theta$ .

Short context: Equiangular sets of lines appear naturally in many areas of mathematics, and the problem of estimating the maximum size of such sets has been studied starting from at least the work of Haantjes in 1948, who proved that $N(3)=N(4)=6$ . In 1973, Lemmens and Seidel formulated a problem of estimating $N_\theta(n)$ for fixed $\theta$ , and proved that $N_\theta(n) = 2n-2$ for $\theta=\arccos \frac{1}{3}$ and sufficiently large n. The Theorem proves a stronger upper bound for all $\theta \neq \arccos \frac{1}{3}$ . This implies that, for all large n, $N_\theta(n)$ is maximised at $\theta=\arccos \frac{1}{3}$ .