# The weakness of the pigeonhole principle under hyperarithmetical reductions

## TL;DR

This paper investigates the infinite pigeonhole principle within computability theory, demonstrating strong cone avoidance for hyperarithmetical reductions and constructing low${}_n$ subsets, advancing understanding of computability aspects of combinatorial principles.

## Contribution

It introduces a new forcing notion that generalizes existing jump control techniques and proves strong cone avoidance results for hyperarithmetical reductions.

## Key findings

- RT^1_2 admits strong cone avoidance for hyperarithmetical reductions
- Existence of low${}_n$ subsets for every Δ^0_n set or its complement
- Answers a question of Wang regarding low${}_n$ subsets

## Abstract

The infinite pigeonhole principle for 2-partitions ($\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.