THE TREE PIGEONHOLE PRINCIPLE IN THE WEIHRAUCH DEGREES

Abstract 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 $\mathsf {TT}^1$ .