• Corpus ID: 119329887

# Definable Combinatorics of Some Borel Equivalence Relations

@article{Chan2017DefinableCO,
title={Definable Combinatorics of Some Borel Equivalence Relations},
author={William Chan and C. G. W. Meehan},
journal={arXiv: Logic},
year={2017}
}
• Published 13 September 2017
• Mathematics
• arXiv: Logic
If $X$ is a set, $E$ is an equivalence relation on $X$, and $n \in \omega$, then define $$[X]^n_E = \{(x_0, ..., x_{n - 1}) \in {}^nX : (\forall i,j)(i \neq j \Rightarrow \neg(x_i \ E \ x_j))\}.$$ For $n \in \omega$, a set $X$ has the $n$-Jonsson property if and only if for every function $f : [X]^n_= \rightarrow X$, there exists some $Y \subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y]^n_=] \neq X$. A set $X$ has the Jonsson property if and only for every function $f : (\bigcup_{n \in… 5 Citations Definable Combinatorics of Graphs and Equivalence Relations Let D = (X, D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0, …, Fn-1: X → X are Borel functions, let DF0, …, Fn-1 be the directed graph MORE DEFINABLE COMBINATORICS AROUND THE FIRST AND SECOND UNCOUNTABLE CARDINAL Assume ZF+AD. The following two continuity results for functions on certain subsets of P(ω1) and P(ω2) will be shown: For every < ω1 and function Φ : [ω1] → ω1, there is a club C ⊆ ω1 and a ζ < so Ordinal definability and combinatorics of equivalence relations ZF + AD proves that the ordinal definable equivalence class of E-class section uniformization is J\'onsson whenever$\kappa\$ is an ordinal.
Definable combinatorics at the first uncountable cardinal
• Mathematics
Transactions of the American Mathematical Society
• 2021
We work throughout in the theory ZF with the axiom of determinacy, AD. We introduce and prove some club uniformization principles under AD and ADR. Using these principles, we establish continuity