Computable vs Descriptive Combinatorics of Local Problems on Trees

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.