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THE TREE PIGEONHOLE PRINCIPLE IN THE WEIHRAUCH DEGREES
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 11 February 2025, pp. 1-23
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- Article
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We study versions of the tree pigeonhole principle,
$\mathsf {TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether 
$\mathsf {TT}^1$ is 
$\Pi ^1_1$-conservative over the ordinary pigeonhole principle, 
$\mathsf {RT}^1$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike 
$\mathsf {RT}^1$, the problem 
$\mathsf {TT}^1$ is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of 
$\mathsf {TT}^1$.
