Where pigeonhole principles meet König lemmas

Belanger, David; Chong, C.; Wang, Wei; Wong, Tin Lok; Yang, Yue

doi:10.1090/tran/8494

Where pigeonhole principles meet König lemmas
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by David Belanger, C. T. Chong, Wei Wang, Tin Lok Wong and Yue Yang PDF

Trans. Amer. Math. Soc. 374 (2021), 8275-8303 Request permission

Abstract:

We study the pigeonhole principle for $\Sigma _2$-definable injections with domain twice as large as the codomain, and the weak König 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.

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