Partition genericity and pigeonhole basis theorems

TL;DR

This paper introduces partition genericity, a new notion in computability theory that combines aspects of genericity and randomness, and demonstrates its relevance to basis theorems and infinite subsets.

Contribution

The paper defines partition genericity, shows its relation to existing notions, and proves that many basis theorems apply to this new concept, answering open questions.

Findings

Every co-hyperimmune set is partition generic.

Every Kurtz random is partition generic.

Partition generic sets admit weak infinite subsets.

Abstract

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.