Borel组合数学中的$Δ^1_1$线性化
$Δ^1_1$ Effectivization in Borel Combinatorics
摘要：
我们开发了一个灵活的方法，证明Borel证人的一些组合属性的$Δ^1_1$对象产生$Δ^1_1$证人。我们使用修改的甘迪-哈林顿力迫法证明二分法，我们可以恢复的复杂性后果，许多已知的二分法与简短和简单的证明。使用我们的方法，我们给出了一个简化的证明，光滑的$Δ^1_1$等价关系是$Δ^1_1$ -可约为相等的;我们证明了Lusin-Novikov和Feldman-Moore定理的有效版本;我们证明了与Hjorth和Miller的二分法有关的新的有效化结果（最初证明使用``forceless，无效，无力”的方法），我们证明了一个新的上界的复杂性的一组Schreier图的$\mathbb{Z}^2$行动。我们还证明了一个等变版本的$G_0$二分法，这意味着这些新的结果和二分法的图引起的博雷尔行动的$\mathbb{Z}^2$。
英文摘要：
We develop a flexible method for showing that Borel witnesses to some combinatorial property of $Δ^1_1$ objects yield $Δ^1_1$ 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 $Δ^1_1$ equivalence relations are $Δ^1_1$-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 $\mathbb{Z}^2$.
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