An official website of the United States government
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$$\mathsf {TT}^1$$.
more »
« less
Free, publicly-accessible full text available February 11, 2026
