# 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} }

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…

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