../

---

Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, Lemma 4.3 (generalized $\Delta$-system), physical p. 5, in the eight-page PDF held by its library source card, Glazer (2026).

Standing. This is an author-recorded reconstruction. It is not an independent review and changes no status and assigns no tier. The following are imported: the Löwenheim--Skolem construction of continuous chains of elementary submodels (T. Jech, Set Theory, third millennium edition, Chapter 12); Fodor's theorem and the stationarity of the set of ordinals below $\omega_2$ of cofinality $\omega_1$ (Jech, Chapter 8); and the cardinal arithmetic $\aleph_1^{\aleph_0}=2^{\aleph_0}$.

Definitions

A family $(A_\xi)_{\xi\in T}$ is a $\Delta$-system with root $R$ if $A_\xi\cap A_\zeta=R$ for all distinct $\xi,\zeta\in T$. Write $S^{\omega_2}_{\omega_1}=\{\xi<\omega_2:\mathrm{cf}(\xi)=\omega_1\}$. For a set $M$, $[M]^{\aleph_0}$ is the set of its countable subsets.

Statement

$$\mathrm{ZFC}+\mathrm{CH}\vdash \text{every family of $\omega_2$ countable sets has a $\Delta$-subsystem of size $\omega_2$}$$

(the source's (4.1)). Precisely: if $\langle S_\alpha:\alpha<\omega_2\rangle$ is a sequence of countable sets, there are a set $T'\subseteq\omega_2$ of size $\omega_2$, an injection $\xi\mapsto\alpha_\xi$ on $T'$, and a countable $R$ with $S_{\alpha_\xi}\cap S_{\alpha_\zeta}=R$ for all distinct $\xi,\zeta\in T'$.

Proof

Enumerate the family as $\langle S_\alpha:\alpha<\omega_2\rangle$. Let $\theta$ be a regular cardinal large enough that the sequence lies in $H(\theta)$. By Löwenheim--Skolem build a chain $\langle M_\xi:\xi<\omega_2\rangle$ of elementary submodels of $H(\theta)$ such that: $|M_\xi|=\omega_1$ and $\omega_1\subseteq M_\xi$; the sequence $\langle S_\alpha\rangle$ belongs to $M_0$; the chain is increasing and continuous ($M_\xi=\bigcup_{\eta<\xi}M_\eta$ at limits $\xi$, a union of an elementary chain being elementary); and $M_{\xi+1}\cap\omega_2$ properly contains $M_\xi\cap\omega_2$. The last clause is arranged at successor steps by putting into $M_{\xi+1}$ an ordinal of $\omega_2\setminus M_\xi$, which exists because $|M_\xi|=\omega_1<\omega_2$.

Two consequences of the setup are used. First, if a countable set $A$ belongs to some $M_\xi$, then $A\subseteq M_\xi$: by elementarity $M_\xi$ contains a surjection $f\colon\omega\to A$ (or $A$ is finite and the same argument applies), and each $f(n)$ is definable in $H(\theta)$ from $f$ and $n\in\omega\subseteq M_\xi$. Second, if $\alpha\in M_\xi$ then $S_\alpha\in M_\xi$, being definable from the sequence and $\alpha$.

For $\xi\in S^{\omega_2}_{\omega_1}$ choose $\alpha_\xi\in(M_{\xi+1}\cap\omega_2)\setminus M_\xi$, put $A_\xi=S_{\alpha_\xi}$, and let $R_\xi=A_\xi\cap M_\xi$. Since $\alpha_\xi\in M_{\xi+1}$, $A_\xi\in M_{\xi+1}$, and $A_\xi$ is countable, so $A_\xi\subseteq M_{\xi+1}$. The map $\xi\mapsto\alpha_\xi$ is injective on $S^{\omega_2}_{\omega_1}$: for $\xi<\zeta$, $\alpha_\xi\in M_{\xi+1}\subseteq M_\zeta$ while $\alpha_\zeta\notin M_\zeta$.

Bounding the roots. Fix $\xi\in S^{\omega_2}_{\omega_1}$. Since $\xi$ is a limit, $M_\xi=\bigcup_{\eta<\xi}M_\eta$, so each of the countably many elements of $R_\xi$ enters the chain at some stage below $\xi$; as $\mathrm{cf}(\xi)=\omega_1$, these countably many stages are bounded by some $\eta(\xi)<\xi$, and $R_\xi\subseteq M_{\eta(\xi)}$.

Fodor. The function $\xi\mapsto\eta(\xi)$ is regressive on the stationary set $S^{\omega_2}_{\omega_1}$, so by Fodor's theorem it is constant, with value $\eta$ say, on a stationary set $T\subseteq S^{\omega_2}_{\omega_1}$; in particular $|T|=\omega_2$. For $\xi\in T$, $R_\xi\in[M_\eta]^{\aleph_0}$.

CH. Since $|M_\eta|=\aleph_1$,

$$\bigl|[M_\eta]^{\aleph_0}\bigr|=\aleph_1^{\aleph_0}=2^{\aleph_0}=\aleph_1$$

under CH. The map $\xi\mapsto R_\xi$ sends the $\omega_2$ elements of $T$ into a set of size $\aleph_1$, so some countable $R$ satisfies $R_\xi=R$ for all $\xi$ in a set $T'\subseteq T$ of size $\omega_2$.

The root. Let $\xi<\zeta$ both lie in $T'$. Then $A_\xi\subseteq M_{\xi+1}\subseteq M_\zeta$, so

$$A_\xi\cap A_\zeta=A_\xi\cap(A_\zeta\cap M_\zeta)=A_\xi\cap R_\zeta =A_\xi\cap R=R,$$

the last step because $R=R_\xi=A_\xi\cap M_\xi\subseteq A_\xi$. Thus $(A_\xi)_{\xi\in T'}=(S_{\alpha_\xi})_{\xi\in T'}$ is a $\Delta$-system of size $\omega_2$ with root $R$.

Boundary. The lemma is applied in Proposition 4.4 to the countable supports of $\omega_2$ names. Those supports need not be pairwise distinct, since two names may share a support even though each support contains its own block $\{\alpha\}\times\omega$; this is why the precise statement above is given for an indexed sequence and returns an injection on indices rather than $\omega_2$ distinct sets. The sequence form is equivalent to the source's family form: a family of $\omega_2$ sets is the injective case, and a sequence with fewer than $\omega_2$ distinct values takes one value $\omega_2$ times, a $\Delta$-system with that value as root.