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Source. E. Glazer, Erdős Problem 501 after adding $\omega_2$ random reals, draft rev10, Theorem 1.1 and Corollary 1.2 (statements, physical p. 1; proof of Theorem 1.1, p. 7; proof of Corollary 1.2 and the counterexample under CH, Section 6, p. 8), in the eight-page PDF held by its library source card, Glazer (2026). Its two inputs are Theorem 3.2 and Theorem 5.1; the counterexample under CH is reconstructed on the CH counterexample page.

Standing. This is an author-recorded reconstruction. It is not an independent review and changes no status and assigns no tier. Imported: the forcing theorem for complete Boolean algebras and the relative consistency it yields, and Gödel's theorem that the constructible universe satisfies ZFC + CH (T. Jech, Set Theory, third millennium edition, Chapters 13--14).

Definitions

$\lambda^*$ is Lebesgue outer measure; $\mathrm{Free}_\omega(\mathcal A)$ and $\mathrm{Prof}(\mathcal A)$ are as on the Theorem 3.2 page. $P$ is the positive assertion of the first question of Problem 501: every family $(A_y)_{y\in\mathbb R}$ of bounded subsets of $\mathbb R$ with $\lambda^*(A_y)<1$ for all $y$ satisfies $\mathrm{Free}_\omega(\mathcal A)$.

Statement

Theorem 1.1. Let $M\models\mathrm{ZFC}+\mathrm{CH}$, let $\kappa=(\omega_2)^M$, and let $G$ be generic over $M$ for the measure algebra $\mathbb B(\kappa\times\omega)$ adding $\kappa$ random reals. In $M[G]$, every family $\mathcal A=(A_y)_{y\in\mathbb R}$ with $\lambda^*(A_y)<1$ for all $y\in\mathbb R$ satisfies $\mathrm{Free}_\omega(\mathcal A)$. Boundedness is not assumed.

Corollary 1.2. If ZFC is consistent, then both $\mathrm{ZFC}+P$ and $\mathrm{ZFC}+\neg P$ are consistent.

Proof of Theorem 1.1

Theorem 5.1 is a theorem of ZFC + CH about the forcing relation, so it holds in $M$: the top condition of $\mathbb B(\kappa\times\omega)$ forces that every family with all outer measures below one has a profile certificate. By the forcing theorem, in $M[G]$ every such family $\mathcal A$ satisfies $\mathrm{Prof}(\mathcal A)$. Theorem 3.2 is a theorem of ZFC, and $M[G]\models\mathrm{ZFC}$, so $\mathrm{Prof}(\mathcal A)\to\mathrm{Free}_\omega(\mathcal A)$ holds in $M[G]$. Hence $M[G]\models\mathrm{Free}_\omega(\mathcal A)$.

Proof of Corollary 1.2

Consistency of $P$. Assume ZFC is consistent, and let $N$ be a model of ZFC. Its constructible universe $L^N$ satisfies ZFC + CH. Adding $\omega_2$ random reals over it, in the sense of Theorem 1.1, yields a model of ZFC in which every family with outer measures below one has an infinite independent set; in particular every family of bounded such sets does, which is $P$. Formally, Theorems 5.1 and 3.2 give $\mathrm{ZFC}+\mathrm{CH}\vdash\mathbb B_{\omega_2}\Vdash P$, and the forcing theorem turns this into $\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})\to\mathrm{Con}(\mathrm{ZFC}+P)$, while $\mathrm{Con}(\mathrm{ZFC})\to\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$ by the constructible universe.

Consistency of $\neg P$. $L^N$ satisfies CH, and CH implies $\neg P$ by the construction on the CH counterexample page: a family of countable, hence null, bounded sets with no infinite independent set. So $L^N\models\mathrm{ZFC}+\neg P$.

Together these give the corollary: $P$ is independent of ZFC relative to $\mathrm{Con}(\mathrm{ZFC})$.

Boundary. The paper's Section 6 separates the argument into formalization units F1--F6; only Lemmas 4.1, 4.2, 4.5, Proposition 4.4 and Theorem 5.1 mention forcing. The library card records that the author's companion Lean development formalizes the independence by a different positive model; nothing on this page bears on that development.