A general lineability criterion for complements of vector spaces

Since the concept of lineability was coined by Gurariy in the early 2000s, only a few results provide a general criterion that guarantees the existence of large algebraic structures. Furthermore, most results in lineability theory are generally positive results, and there are a small amount of negative results. In this paper, we provide a non-constructive general ( α , β ) -lineability criterion in the context of complements of vector spaces, and we present two criteria of non- ( α , β ) -spaceability. Several applications are presented.