Where Pigeonhole Principles meet K\"onig Lemmas

TL;DR

This paper explores the relationships between pigeonhole principles and K"onig lemmas within computability theory, demonstrating implications for randomness and logical strength of certain combinatorial principles.

Contribution

It establishes that a weak K"onig lemma variant implies the existence of 2-random reals and is conservative over a specific pigeonhole principle, revealing their relative logical strengths.

Findings

Weak K"onig lemma implies existence of 2-random reals

Weak K"onig lemma is conservative over a specific pigeonhole principle

The pigeonhole principle studied is strictly weaker than the usual one for $oldsymbol{ m extSigma_2}$-definable injections

Abstract

We study the pigeonhole principle for $\Sigma_2$-definable injections with domain twice as large as the codomain, and the weak K\"onig lemma for $\Delta^0_2$-definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of $2$-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for $\Sigma_2$-definable injections.