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The tree pigeonhole principle in the Weihrauch degrees

Damir Dzhafarov, Reed Solomon, Manlio Valenti

TL;DR

The authors analyze the tree pigeonhole principle $\mathsf{TT}^1$ within the Weihrauch framework, relating it to the infinitary pigeonhole principle $\mathsf{RT}^1$. They introduce the notion of the first-order part $^1\mathsf{P}$ and show that the first-order part of $\mathsf{TT}^1$ precisely coincides with $\mathsf{RT}^1$, yielding a positive analogue to a well-known question from reverse mathematics. They further demonstrate that $\mathsf{TT}^1$ is not equivalent to any first-order statement, and establish a sharp separation between bounded-color and unbounded-color versions via a series of intricate reductions. A central technical contribution is a new combinatorial framework (dense-or-cone variants and rake constructions) that enables the reduction of $^1\mathsf{TT}^1_{\mathbb{N}}$ to $\mathsf{RT}^1_{\mathbb{N}}$, clarifying the strength and limits of $\mathsf{TT}^1$ in the Weihrauch degrees and aligning the picture with analogous reverse-mathematical insights.

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. Two outstanding questions from the latter investigation are whether $\mathsf{TT}^1$ is $Π^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf{RT}^1$, and whether it is equivalent to any first-order statement of second-order arithmetic. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate, and answer in the affirmative, the analogue of the first question for Weihrauch reducibility. We then use this, in combination with other results, to answer in the negative the analogue of the second question. Our proofs develop a new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf{TT}^1$.

The tree pigeonhole principle in the Weihrauch degrees

TL;DR

The authors analyze the tree pigeonhole principle within the Weihrauch framework, relating it to the infinitary pigeonhole principle . They introduce the notion of the first-order part and show that the first-order part of precisely coincides with , yielding a positive analogue to a well-known question from reverse mathematics. They further demonstrate that is not equivalent to any first-order statement, and establish a sharp separation between bounded-color and unbounded-color versions via a series of intricate reductions. A central technical contribution is a new combinatorial framework (dense-or-cone variants and rake constructions) that enables the reduction of to , clarifying the strength and limits of in the Weihrauch degrees and aligning the picture with analogous reverse-mathematical insights.

Abstract

We study versions of the tree pigeonhole principle, , in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics. Two outstanding questions from the latter investigation are whether is -conservative over the ordinary pigeonhole principle, , and whether it is equivalent to any first-order statement of second-order arithmetic. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate, and answer in the affirmative, the analogue of the first question for Weihrauch reducibility. We then use this, in combination with other results, to answer in the negative the analogue of the second question. Our proofs develop a new combinatorial machinery for constructing and understanding solutions to instances of .
Paper Structure (6 sections, 19 theorems, 6 equations, 2 figures)

This paper contains 6 sections, 19 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The complexity of building monochromatic solutions
  4. Bounded number of colors
  5. Unbounded number of colors
  6. Questions

Key Result

Theorem 3.2

For all $k \geq 2$, $\mathsf{TT}^1_k$-$\mathsf{Ext} \equiv_{\text{\upshape W}} \mathsf{WF}$.

Figures (2)

  • Figure 1: An illustration of a $C$-rake with $C = \{ 0,1,2 \}$ of height $9$. The color under $f$ of the nodes in blocks $0$ and $1$ is indicated.
  • Figure 2: A $C$-rake with $|C|=3$ (represented by hollow nodes, connected by interrupted lines), and a set $S \trianglelefteq R$ of height $3$ (represented by solid nodes, connected by uninterrupted lines).

Theorems & Definitions (46)

  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 36 more