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Source. S. Lee, Relative independence of Erdős problem #501, second version dated 2026-06-01, Theorem 1.1 and Corollary 1.2 (statements, physical p. 1; proof of Theorem 1.1, Section 2, pp. 2--3), in the six-page PDF held by its library source card, Lee (2026). Its input is Lemma 2.1; the negative half of the corollary is 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 for the corollary: the equiconsistency of FMEA with a measurable cardinal, which the source cites to D. H. Fremlin, Real-valued-measurable cardinals, version of 19 September 2009, 1D(e) and 2E; and Gödel's theorem that the constructible universe satisfies CH.

Definitions

$m$ and $m^*$ are Lebesgue measure and outer measure. FMEA, the Full Measure Extension Axiom, asserts that there is a countably additive measure $\nu\colon\mathcal P(\mathbb R)\to[0,\infty]$ extending Lebesgue measure. $P$ 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 $m^*(A_y)<1$ admits an infinite independent set, an infinite $X\subseteq\mathbb R$ with $x\notin A_y$ for all distinct $x,y\in X$. For a family $(A_y)$ and $x\in\mathbb R$, $B_x=\{y:x\in A_y\}$. $\mathbb R^{<\omega}$ is the set of finite sequences of reals, a sequence $s$ of length $|s|=n$ being a function on $\{0,\dots,n-1\}$; $f\restriction n$ is the restriction of $f\colon\omega\to\mathbb R$ to $\{0,\dots,n-1\}$.

Statement

Theorem 1.1. Under ZFC + FMEA, whenever each $A_y\subseteq\mathbb R$ ($y\in\mathbb R$) has outer measure $m^*(A_y)<1$, some infinite $X\subseteq\mathbb R$ is independent for the family. Boundedness is not assumed, so the theorem implies $P$ under FMEA.

Corollary 1.2. Assuming $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$, ZFC neither proves nor refutes $P$; since FMEA is equiconsistent with a measurable cardinal, the consistency of ZFC plus a measurable cardinal already suffices.

Proof of Theorem 1.1

Fix a measure $\nu\colon\mathcal P(\mathbb R)\to[0,\infty]$ extending Lebesgue measure, a family $(A_y)$ with $m^*(A_y)<1$ for every $y$, a well-ordering $\preceq$ of $\mathbb R$ and a point $r_0\in\mathbb R$.

The pools. For $s\in\mathbb R^{<\omega}$ define

$$C_s=\mathbb R\setminus\bigcup_{i<|s|} \bigl(A_{s(i)}\cup B_{s(i)}\cup\{s(i)\}\bigr)$$

(the source's (2)) and

$$Q_s=\{a\in C_s:\nu(C_s\setminus B_a)=\infty\}$$

(the source's (3)). Let $G(s)$ be the $\preceq$-least element of $Q_s$ if $Q_s\neq\varnothing$, and $r_0$ otherwise. By recursion on $\omega$ there is $f\colon\omega\to\mathbb R$ with $f(n)=G(f\restriction n)$ for every $n$.

Infinite measure is preserved. We show by induction that $\nu(C_{f\restriction n})=\infty$ for every $n$. For $n=0$, $f\restriction0$ is the empty sequence and $C_{f\restriction0}=\mathbb R$, of infinite $\nu$-measure since $\nu$ extends Lebesgue measure. Suppose $\nu(C_{f\restriction n})=\infty$. Lemma 2.1 applied to $C=C_{f\restriction n}$ gives an $a\in C_{f\restriction n}$ with $\nu(C_{f\restriction n}\setminus B_a)=\infty$, so $Q_{f\restriction n}\neq\varnothing$ and $f(n)=G(f\restriction n)$ is its $\preceq$-least element; thus $f(n)\in C_{f\restriction n}$ and

$$\nu(C_{f\restriction n}\setminus B_{f(n)})=\infty.$$

By the definition of the pools,

$$C_{f\restriction(n+1)} =C_{f\restriction n}\setminus\bigl(A_{f(n)}\cup B_{f(n)}\cup\{f(n)\}\bigr) =\bigl(C_{f\restriction n}\setminus B_{f(n)}\bigr)\setminus \bigl(A_{f(n)}\cup\{f(n)\}\bigr).$$

By the comparison $\nu\le m^*$ recorded on the Lemma 2.1 page, $\nu(A_{f(n)})\le m^*(A_{f(n)})<1$, and $\nu(\{f(n)\})=0$ because $\nu$ extends Lebesgue measure; so $\nu(A_{f(n)}\cup\{f(n)\})<\infty$. Removing a set of finite measure from a set of infinite measure leaves infinite measure, so $\nu(C_{f\restriction(n+1)})=\infty$.

The independent set. Put $X=\{f(n):n<\omega\}$. Let $i<j$. The pools decrease along $f$, so

$$f(j)\in C_{f\restriction j}\subseteq C_{f\restriction(i+1)} =C_{f\restriction i}\setminus\bigl(A_{f(i)}\cup B_{f(i)}\cup\{f(i)\}\bigr).$$

Hence $f(j)\neq f(i)$, $f(j)\notin A_{f(i)}$, and $f(j)\notin B_{f(i)}$; the last says $f(i)\notin A_{f(j)}$. So $X$ is infinite, and $x\notin A_y$ for all distinct $x,y\in X$.

Proof of Corollary 1.2

Theorem 1.1 gives $\mathrm{ZFC}+\mathrm{FMEA}\vdash P$, so $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$ implies $\mathrm{Con}(\mathrm{ZFC}+P)$. It also implies $\mathrm{Con}(\mathrm{ZFC})$, hence $\mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$ through the constructible universe, and CH implies $\neg P$ by the CH counterexample; so $\mathrm{Con}(\mathrm{ZFC}+\neg P)$. Together, neither $P$ nor $\neg P$ is provable in ZFC. The second sentence of the corollary follows from the imported equiconsistency: the consistency of a measurable cardinal gives $\mathrm{Con}(\mathrm{ZFC}+\mathrm{FMEA})$.

Boundary. FMEA is used only to have $\nu$ at all; the recursion itself is elementary once Lemma 2.1 is available. The measure-extension hypothesis is what Glazer's argument, on the Glazer Theorem 1.1 page, removes by forcing.