Partition genericity and pigeonhole basis theorems

Benoit Monin (1) , Ludovic Patey (2, 3)

1 LACL - Laboratoire d'Algorithmique Complexité et Logique
2 AGL - Algèbre, géométrie, logique
3 CNRS - Centre National de la Recherche Scientifique

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.

Domaines: Logique [math.LO]

Benoit Monin, Ludovic Patey. Partition genericity and pigeonhole basis theorems. The Journal of Symbolic Logic, inPress, ⟨10.1017/jsl.2022.69⟩. ⟨hal-03790535⟩