$\Delta^1_1$ Effectivization in Borel Combinatorics

Abstract

We develop a flexible method for showing that Borel witnesses to some combinatorial property of $\Delta^1_1$ objects yield $\Delta^1_1$ witnesses. We use a modification the Gandy--Harrington forcing method of proving dichotomies, and we can recover the complexity consequences of many known dichotomies with short and simple proofs. Using our methods, we give a simplified proof that smooth $\Delta^1_1$ equivalence relations are $\Delta^1_1$-reducible to equality; we prove effective versions of the Lusin--Novikov and Feldman--Moore theorems; we prove new effectivization results related to dichotomy theorems due to Hjorth and Miller (originally proven using ``forceless, ineffective, and powerless" methods); and we prove a new upper bound on the complexity of the set of Schreier graphs for $\mathbb{Z}^2$ actions. We also prove an equivariant version of the $G_0$ dichotomy that implies some of these new results and a dichotomy for graphs induced by Borel actions of $\mathbb{Z}^2$.