Additivity and lineability in vector spaces

Gámez-Merino, Munoz-Fernández and Seoane-Sepúlveda proved that if additivity A(F)>c, then F is A(F)-lineable where F⊆RR. They asked if A(F)>c can be weakened. We answer this question in negative. Moreover, we introduce and study the notions of homogeneous lineability number and lineability number of subsets of linear spaces.