Additivity and lineability in vector spaces

Gámez-Merino, Munoz-Fernández and Seoane-Sepúlveda proved that if additivity \(\mathcal A(\mathcal F)>\mathfrak c\), then \(\mathcal F\) is \(\mathcal A(\mathcal F)\)-lineable where \(\mathcal F\subseteq\mathbb R^\mathbb R\). They asked if \(\mathcal A(\mathcal F)>\mathfrak c\) 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.