Additivity and lineability in vector spaces

TL;DR

This paper investigates the conditions under which certain subsets of real-valued functions are lineable, confirming that the previously established additivity threshold cannot be lowered, and introduces new concepts of homogeneous lineability number.

Contribution

It proves that the additivity threshold for lineability cannot be weakened and introduces the concepts of homogeneous lineability number and lineability number.

Findings

The additivity threshold cannot be lowered below the continuum for lineability.

Introduces and studies the notions of homogeneous lineability number and lineability number.

Confirms the optimality of previous lineability results.

Abstract

G\'amez-Merino, Munoz-Fern\'andez and Seoane-Sep\'ulveda 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.