2025/04/15 by Bachoc, Christine, Bekker, Bram, Moustrou, Philippe +1
#51K99 #52C10 #90C22 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2504.11086
For t ∈ [-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 α(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 α(n, t) ≤ 2(n+1) for all n and t ≤ 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 ≤ 5 and to enumerate all possible constructions for n ≤ 3 and t ≤ 0.