  We study the sizes of delta-additive sets of unit vectors in a d-dimensional
normed space: the sum of any two vectors has norm at most delta. One-additive
sets originate in finding upper bounds of vertex degrees of Steiner Minimum
Trees in finite dimensional smooth normed spaces (Z. F\"uredi, J. C. Lagarias,
F. Morgan, 1991). We show that the maximum size of a delta-additive set over
all normed spaces of dimension d grows exponentially in d for fixed delta>2/3,
stays bounded for delta<2/3, and grows linearly at the threshold delta=2/3.
Furthermore, the maximum size of a 2/3-additive set in d-dimensional normed
space has the sharp upper bound of d, with the single exception of spaces
isometric to three-dimensional l^1 space, where there exists a 2/3-additive set
of four unit vectors.
