Computable vs Descriptive Combinatorics of Local Problems on Trees

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 $\mathrm{\Delta }$-regular trees, $\mathrm{\Delta }\in \omega$. We show that such a problem admits a computable solution on every highly computable $\mathrm{\Delta }$-regular forest if and only if it admits a Baire measurable solution on every Borel $\mathrm{\Delta }$-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $\mathrm{\Delta }$ forest then it admits a continuous solution on every maximum degree $\mathrm{\Delta }$ Borel graph with appropriate topological hypotheses, though the converse does not hold.