  There exist two notions of typicality in computability theory, namely,
genericity and randomness. In this article, we introduce a new notion of
genericity, called partition genericity, which is at the intersection of these
two notions of typicality, and show that many basis theorems apply to partition
genericity. More precisely, we prove that every co-hyperimmune set and every
Kurtz random is partition generic, and that every partition generic set admits
weak infinite subsets. In particular, we answer a question of Kjos-Hanssen and
Liu by showing that every Kurtz random admits an infinite subset which does not
compute any set of positive Hausdorff dimension. Partition genericty is a
partition regular notion, so these results imply many existing pigeonhole basis
theorems.
