Definable Combinatorics of Some Borel Equivalence Relations
Definable Combinatorics of Some Borel Equivalence Relations
复制标题
一些Borel等价关系的可定义组合
DOI：
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发表时间：
2017
期刊：
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通讯作者：
C. Meehan
中科院分区：
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作者：
William Chan;C. Meehan
如果\(X\)是一个集合，\(E\)是\(X\)上的一个等价关系，并且\(n\in\omega\)，那么定义
\([X]^n_E =\{(x_0,\ldots,x_{n - 1})\in {}^nX:(\forall i,j)(i\neq j\Rightarrow\neg(x_i E x_j))\}\)。
对于\(n\in\omega\)，一个集合\(X\)具有\(n\)-约恩松性质当且仅当对于每一个函数\(f:[X]^n_=\to X\)，存在某个\(Y\subseteq X\)使得\(X\)与\(Y\)一一对应，并且\(f[[Y]^n_=]\neq X\)。一个集合\(X\)具有约恩松性质当且仅当对于每一个函数\(f:(\bigcup_{n\in\omega}[X]^n_=)\to X\)，存在某个\(Y\subseteq X\)使得\(X\)与\(Y\)一一对应，并且\(f[\bigcup_{n\in\omega}[Y]^n_=]\neq X\)。
设\(n\in\omega\)，\(X\)是一个波兰空间，并且\(E\)是\(X\)上的一个等价关系。\(E\)具有\(n\)-米谢尔斯基性质当且仅当对于所有余密的\(C\subseteq {}^nX\)，存在某个\(\Delta_1^1\)的\(A\subseteq X\)使得\(E\leq_{\Delta_1^1}E\upharpoonright A\)并且\([A]^n_E\subseteq C\)。
将考虑以下等价关系：\(E_0\)定义在\({}^\omega2\)上，\(x E_0 y\)当且仅当\((\exists n)(\forall k > n)(x(k)=y(k))\)。\(E_1\)定义在\({}^\omega({}^\omega2)\)上，\(x E_1 y\)当且仅当\((\exists n)(\forall k > n)(x(k)=y(k))\)。\(E_2\)定义在\({}^\omega2\)上，\(x E_2 y\)当且仅当\(\sum\{\frac{1}{n + 1}:n\in x\triangle y\}<\infty\)，其中\(\triangle\)表示对称差。\(E_3\)定义在\({}^\omega({}^\omega2)\)上，\(x E_3 y\)当且仅当\((\forall n)(x(n)E_0 y(n))\)。
霍尔肖瑟和杰克逊已经表明在\(\mathsf{AD}\)下\(\mathbb{R}\)是约恩松的。将会表明\(E_0\)不具有\(3\)-米谢尔斯基性质，并且\(E_1\)、\(E_2\)和\(E_3\)不具有\(2\)-米谢尔斯基性质。在\(\mathsf{ZF + AD}\)下，\({}^\omega2/E_0\)不具有\(3\)-约恩松性质。
If $X$ is a set, $E$ is an equivalence relation on $X$, and $n in omega$, then define $$[X]^n_E = {(x_0, ..., x_{n - 1}) in {}^nX : (forall i,j)(i
eq j Rightarrow
eg(x_i E x_j))}.$$
For $n in omega$, a set $X$ has the $n$-Jonsson property if and only if for every function $f : [X]^n_=
ightarrow X$, there exists some $Y subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y]^n_=]
eq X$. A set $X$ has the Jonsson property if and only for every function $f : (igcup_{n in omega}[X]^n_=)
ightarrow X$, there exists some $Y subseteq X$ with $X$ and $Y$ in bijection so that $f[igcup_{n in omega} [Y]^n_=]
eq X$.
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 $sum{frac{1}{n + 1} : n in x riangle y} < infty$, where $ riangle$ 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 Jonsson 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 $3$-Jonsson property.