Obtuse almost-equiangular sets

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 Lovász 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 $\alpha (n,t)\le 2(n+1)$ for all $n$ and $t\le 0$, with equality only at $t=-1\mathrm{/}n$. This allows us to show the uniqueness of the optimal construction at $t=-1\mathrm{/}n$ for $n\le 5$ and to enumerate all possible constructions for $n\le 3$ and $t\le 0$.