Definable Combinatorics of Some Borel Equivalence Relations

Abstract

If $X$ is a set, $E$ is an equivalence relation on $X$, and $n\in \omega$, then define $$[X{]}_E^n=\{(x_0,...,x_{n-1})\in {}^{n}X:(\forall i,j)(i\ne j\Rightarrow \neg (x_{i}E{x}_j))\}.$$ For $n\in \omega$, a set $X$ has the $n$-Jónsson property if and only if for every function $f:[X{]}_{=}^n\to X$, there exists some $Y\subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y{]}_{=}^n]\ne X$. A set $X$ has the Jónsson property if and only for every function $f:({\bigcup }_{n\in \omega }[X{]}_{=}^n)\to X$, there exists some $Y\subseteq X$ with $X$ and $Y$ in bijection so that $f[{\bigcup }_{n\in \omega }[Y{]}_{=}^n]\ne X$. Let $n\in \omega$, $X$ be a Polish space, and $E$ be an equivalence relation on $X$. $E$ has the $n$-Mycielski property if and only if for all comeager $C\subseteq {}^{n}X$, there is some ${\mathbf{\Delta }}_1^1$ $A\subseteq X$ so that $E{\le }_{{\mathbf{\Delta }}_1^1}E\upharpoonright A$ and $[A{]}_E^n\subseteq C$. The following equivalence relations will be considered: $E_0$ is defined on ${}^{\omega }2$ by $x{E}_{0}y$ if and only if $(\exists n)(\forall k>n)(x(k)=y(k))$. $E_1$ is defined on ${}^{\omega }({}^{\omega }2)$ by $x{E}_{1}y$ if and only if $(\exists n)(\forall k>n)(x(k)=y(k))$. $E_2$ is defined on ${}^{\omega }2$ by $x{E}_{2}y$ if and only if $\sum \{\frac{1}{n+1}:n\in x△y\}<\infty$, where $△$ denotes the symmetric difference. $E_3$ is defined on ${}^{\omega }({}^{\omega }2)$ by $x{E}_{3}y$ if and only if $(\forall n)(x(n)E_{0}y(n))$. Holshouser and Jackson have shown that $\mathbb{R}$ is Jónsson under $\mathsf{AD}$. It will be shown that $E_0$ does not have the $3$-Mycielski property and that $E_1$, $E_2$, and $E_3$ do not have the $2$-Mycielski property. Under $\mathsf{ZF}+\mathsf{AD}$, ${}^{\omega }2/E_0$ does not have the $3$-Jónsson property.