We discuss a notion of forcing that characterizes enumeration -genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration -generic sets and their degrees. We construct an enumeration operator such that, for any , the set is enumeration -generic and has the same jump complexity as . We deduce from this and other recent results from the literature that not only does every degree bound an enumeration -generic degree such that , but also that, if is nonzero, then we can find such satisfying . We conclude by proving the existence of both a nonzero low and a properly nonsplittable enumeration -generic degree, hence proving that the class of -generic degrees is properly subsumed by the class of enumeration -generic degrees.
"Enumeration -Genericity in the Local Enumeration Degrees." Notre Dame J. Formal Logic 59 (4) 461 - 489, 2018. https://doi.org/10.1215/00294527-2018-0008