$\Delta^1_1$ Effectivization in Borel Combinatorics

TL;DR

This paper introduces a new method for deriving effective descriptive set theory results in Borel combinatorics, simplifying proofs and extending known theorems to the $ extit{effective}$ setting.

Contribution

It develops a flexible technique based on Gandy--Harrington forcing to effectively analyze Borel combinatorial properties of $ extit{Δ}^1_1$ objects, leading to new and simplified results.

Findings

Simplified proof that smooth $ extit{Δ}^1_1$ equivalence relations are $ extit{Δ}^1_1$-reducible to equality

Effective versions of the Lusin--Novikov and Feldman--Moore theorems

New upper bounds on the complexity of Schreier graphs for $ extit{Z}^2$ actions

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 $\mathbb{Z}^2$.