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Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, Proposition 4.4 (homogeneous Borel reading), physical pp. 5--6, in the eight-page PDF held by its library source card, Glazer (2026). It uses Lemma 4.1 and Lemma 4.3.

Standing. This is an author-recorded reconstruction. It is not an independent review and changes no status and assigns no tier. Imported: the conventions (R1)--(R5) on the Lemma 4.1 page, and the fact that there are at most $2^{\aleph_0}$ Borel maps between two standard Borel spaces (a Borel map is determined by the preimages of a countable base, and there are $2^{\aleph_0}$ Borel sets; A. S. Kechris, Classical Descriptive Set Theory, Chapter 11).

Definitions

Let $\kappa=\omega_2$, $\Theta=\kappa\times\omega$, and $D_\alpha=\{\alpha\}\times\omega$ for $\alpha<\kappa$; the blocks $D_\alpha$ partition $\Theta$. For a bijection $\pi\colon P\to P'$ between coordinate sets and $v\in2^{P'}$, write $v\circ\pi\in2^P$ for the pullback; the source writes it $\pi^{-1}(v)$. For a pair $(P',D')$ with $D'\subseteq P'$ countable and a fixed enumeration of $D'$, its isomorphism type is determined by the cardinality of $P'\setminus D'$, one of $0,1,2,\dots,\aleph_0$; two pairs of the same type admit a bijection matching the enumerations of the distinguished subsets.

Statement

The following is provable in ZFC + CH. Let $p\in\mathbb B(\Theta)$, let $X$ be a standard Borel space, and for each $\alpha<\kappa$ let $\dot w_\alpha$ be a name for an element of $X$. There are

• a set $J\subseteq\kappa$ of size $\kappa$;
• a countable root $R$ supporting $p$;
• pairwise disjoint countable petals $P_\alpha$ with $D_\alpha\subseteq P_\alpha$, for $\alpha\in J$;
• a fixed countable pair $D\subseteq P$ with an enumeration $\langle d_n:n<\omega\rangle$ of $D$, and bijections $\pi_\alpha\colon P\to P_\alpha$ with $\pi_\alpha(d_n)=(\alpha,n)$;
• a single Borel map $F\colon2^R\times2^P\to X$

such that

$$\Vdash_{\mathbb B(\Theta)}\dot w_\alpha =F\bigl(\dot G\restriction R,\ (\dot G\restriction P_\alpha)\circ\pi_\alpha\bigr) \qquad(\alpha\in J)$$

(the source's (4.2)).

Proof

Supports. By (R1) choose a countable $R_0\subseteq\Theta$ supporting $p$. By Lemma 4.1, for each $\alpha<\kappa$ choose a countable $S_\alpha\subseteq\Theta$ reading $\dot w_\alpha$ through a Borel map $F_\alpha\colon2^{S_\alpha}\to X$, and enlarge $S_\alpha$ to contain $R_0\cup D_\alpha$ (a reading survives enlargement, as noted on the Lemma 4.1 page). The sets $S_\alpha$ are countable. They need not be pairwise distinct, since two names may share a support: Lemma 4.3 as reconstructed applies to the sequence $\langle S_\alpha:\alpha<\kappa\rangle$ as it stands, and a member repeated inside the $\Delta$-subsystem below equals the root $R$, so its block lies in $R$ and its index is discarded under "Blocks inside petals".

Delta-system. Apply Lemma 4.3 to $\langle S_\alpha:\alpha<\kappa\rangle$: there are $J_0\subseteq\kappa$ of size $\kappa$ and a countable $R$ with $S_\alpha\cap S_\beta=R$ for distinct $\alpha,\beta\in J_0$. Since $R_0\subseteq S_\alpha$ for all $\alpha$, $R_0\subseteq R$, so $R$ supports $p$. Put $P_\alpha=S_\alpha\setminus R$ for $\alpha\in J_0$; for distinct $\alpha,\beta\in J_0$, $P_\alpha\cap P_\beta=(S_\alpha\cap S_\beta)\setminus R=\varnothing$.

Blocks inside petals. The countable set $R$ meets only countably many of the pairwise disjoint blocks $D_\alpha$. Let $J_1$ be $J_0$ with those indices removed; $|J_1|=\kappa$, and for $\alpha\in J_1$, $D_\alpha\cap R=\varnothing$, so $D_\alpha\subseteq S_\alpha\setminus R=P_\alpha$.

One isomorphism type. The pairs $(P_\alpha,D_\alpha)$, with $D_\alpha$ enumerated as $\langle(\alpha,n):n<\omega\rangle$, fall into countably many isomorphism types. A set of size $\omega_2$ is not a countable union of sets of size at most $\omega_1$, so some type contains $\kappa$ many indices; let $J_2\subseteq J_1$ be those indices. Fix a countable pair $D\subseteq P$ of that type with an enumeration $\langle d_n\rangle$ of $D$, and for $\alpha\in J_2$ fix a bijection $\pi_\alpha\colon P\to P_\alpha$ with $\pi_\alpha(d_n)=(\alpha,n)$.

Pulling back. For $\alpha\in J_2$, $S_\alpha=R\sqcup P_\alpha$, so $2^{S_\alpha}$ is identified with $2^R\times2^{P_\alpha}$. Define $\tilde F_\alpha\colon2^R\times2^P\to X$ by

$$\tilde F_\alpha(u,v)=F_\alpha\bigl(u\cup(v\circ\pi_\alpha^{-1})\bigr),$$

a Borel map, being $F_\alpha$ composed with the homeomorphism $(u,v)\mapsto u\cup(v\circ\pi_\alpha^{-1})$ of $2^R\times2^P$ onto $2^{S_\alpha}$. Since $\dot G\restriction S_\alpha=(\dot G\restriction R)\cup(\dot G\restriction P_\alpha)$ and $\dot G\restriction P_\alpha=((\dot G\restriction P_\alpha)\circ\pi_\alpha)\circ\pi_\alpha^{-1}$,

$$\Vdash\dot w_\alpha=F_\alpha(\dot G\restriction S_\alpha) =\tilde F_\alpha\bigl(\dot G\restriction R, (\dot G\restriction P_\alpha)\circ\pi_\alpha\bigr).$$

Counting. The maps $\tilde F_\alpha$ ($\alpha\in J_2$) all belong to the set of Borel maps $2^R\times2^P\to X$, which has size at most $2^{\aleph_0}=\aleph_1$ under CH. Since $|J_2|=\aleph_2$, some Borel map $F$ equals $\tilde F_\alpha$ for every $\alpha$ in a set $J\subseteq J_2$ of size $\kappa$. This $J$, $R$, $(P_\alpha)_{\alpha\in J}$, $D\subseteq P$, $(\pi_\alpha)_{\alpha\in J}$ and $F$ satisfy the statement.

Boundary. CH enters twice: through Lemma 4.3 and through the count of Borel maps. The proposition is applied in Theorem 5.1 with $X=\mathcal O^{\mathbb Z}$.