Definable Combinatorics of Some Borel Equivalence Relations

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 \(n\)-Jónsson 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 Jónsson property if and only for every function \(f : (\bigcup_{n \in \omega}[X]^n_=) \rightarrow X\), there exists some \(Y \subseteq X\) with \(X\) and \(Y\) in bijection so that \(f[\bigcup_{n \in \omega} [Y]^n_=] \neq X\). 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 \(C \subseteq {}^nX\), there is some \(\mathbf{\Delta_1^1}\) \(A \subseteq X\) so that \(E \leq_{\mathbf{\Delta_1^1}} E \upharpoonright A\) and \([A]^n_E \subseteq C\). The following equivalence relations will be considered: \(E_0\) is defined on \({}^\omega2\) by \(x \ E_0 \ y\) if and only if \((\exists n)(\forall k > n)(x(k) = y(k))\). \(E_1\) is defined on \({}^\omega({}^\omega2)\) by \(x \ E_1 \ y\) if and only if \((\exists n)(\forall k > n)(x(k) = y(k))\). \(E_2\) is defined on \({}^\omega2\) by \(x \ E_2 \ y\) if and only if \(\sum\{\frac{1}{n + 1} : n \in x \ \triangle \ y\} < \infty\), where \(\triangle\) denotes the symmetric difference. \(E_3\) is defined on \({}^\omega({}^\omega2)\) 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 \(\mathsf{ZF + AD}\), \({}^\omega 2 / E_0\) does not have the \(3\)-Jónsson property.