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arXiv · 2504.11086

Obtuse almost-equiangular sets

Christine Bachoc·Bram Bekker·Philippe Moustrou·Fernando Mário de Oliveira Filho
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Abstract

For $t \in [-1, 1)$, a set of points on the $(n-1)$-dimensional unit sphere is called $t$-almost equiangular if among any three distinct points there is a pair with inner product $t$. We propose a semidefinite programming upper bound for the maximum cardinality $\alpha(n, t)$ of such a set based on an extension of the Lov\'asz theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of $n$ and $t$ it is better. We also refine existing spectral methods to show that $\alpha(n, t) \leq 2(n+1)$ for all $n$ and $t \leq 0$, with equality only at $t = -1/n$. This allows us to show the uniqueness of the optimal construction at $t = -1/n$ for $n \leq 5$ and to enumerate all possible constructions for $n \leq 3$ and $t \leq 0$.

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Christine Bachoc, Bram Bekker, Philippe Moustrou, Fernando Mário de Oliveira Filho. 2025-04-15. Obtuse almost-equiangular sets. https://arxiv.org/abs/2504.11086

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