Computable vs Descriptive Combinatorics of Local Problems on Trees

TL;DR

This paper explores the relationship between computable and descriptive solutions for local problems on regular trees, establishing equivalences and implications between computability, Baire measurability, and continuity under various conditions.

Contribution

It characterizes when computable solutions exist in terms of Baire measurable and continuous solutions on different classes of regular forests and graphs.

Findings

Computable solutions on highly computable forests correspond to Baire measurable solutions on Borel forests.

Existence of computable solutions on maximum degree forests implies the existence of continuous solutions on Borel graphs.

The converse of the above implication does not hold.

Abstract

We study the position of the computable setting in the "common theory of locality" developed in arXiv:2106.02066 and arXiv:2204.09329 for local problems on $\Delta$-regular trees, $\Delta \in \omega$. We show that such a problem admits a computable solution on every highly computable $\Delta$-regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta$-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $\Delta$ forest then it admits a continuous solution on every maximum degree $\Delta$ Borel graph with appropriate topological hypotheses, though the converse does not hold.