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In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition.

(EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of sets of rationally independent real numbers; then $\bigcup_{n \in \omega} M_n$ has inner measure zero.

Their Theorem 2 establishes the equivalence of CH with the assertion that $\mathbb{R} = \bigcup_{n \in \omega} M_n$ for some family $\lbrace M_n : n \in \omega \rbrace$ of sets of rationally independent numbers.

Has (EK*) ever been settled?

The question is prompted by the related question (and its discussion) monochromatic cycle-free colouring of the complete graph on R?

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Addendum: P. Komjath has pointed out that the following argument is originally due to Erdos and Kunen. See page 136 here.

Claim 1: (Mycielski) Suppose $A$ is a compact subset of plane of positive area. Then there exist perfect sets $P, Q$ such that $Q$ has positive length and $P \times Q \subseteq A$.

Proof: (Martin) Let $M$ be a countable transitive model of ZFC containing a code of $A$. Let $N$ be a forcing extension of $M$ in which continuum is at least $\omega_3$ and Martin's axiom (MA) holds.

Claim 2: Assume MA + $2^{\omega} \geq \omega_3$. For every collection $\{K_{\alpha} : \alpha < \omega_2\}$ of compact sets of positive measure, there is a compact set $ K$ of positive measure such that $ K \subseteq K_{\alpha}$ for at least $ \omega_2$ many $ \alpha$'s.

Proof: WLOG, we may assume that each $K_{\alpha} \subseteq [0, 1]$ with $\mu(K_{\alpha}) > 0.5$. Let $U_{\alpha} = [0, 1] \backslash K_{\alpha}$. Consider the poset $P$ consisting of open subsets of $[0, 1]$ of measure strictly less than $0.5$ ordered by reverse inclusion; i.e., $U \leq V$ iff $V \subseteq U$. $P$ has ccc because the measure algebra is separable. By MA and $2^{\omega} \geq \omega_3$, it follows that every subset of $P$ of size $\omega_2$ has a centered subset of size $\omega_2$. Hence for some $S \in [\omega_2]^{\omega_2}$, any finite union of sets in $\{U_{\alpha}: \alpha \in S\}$ is in $P$. Claim 2 follows.

The claim implies that, in $N$, there is a compact set $Q$ of positive measure such that $W = \{x: Q \subseteq A_x = \{y: (x, y) \in A\}\}$ has size at least $\omega_2$. Since $A$ is closed, $W$ is a coanalytic set hence it is a union of $\omega_1$ many Borel sets. Thus $W$ must contain a prefect set $P$. So $P \times Q \subseteq A$.

Now the statement $(\exists P, Q)(P, Q \text{ are perfect}, Q \text{ has positive measure and }P \times Q \subseteq A)$ is $\Sigma^{1}_2(A)$ so by Shoenfield's absoluteness theorem it holds in $M$ and hence, by $\Pi_1^1$ absoluteness, also in $V$.

Claim 3: Assume $2^{\omega} \geq \omega_2$. Let $X_n$, for $n < \omega$, be pairwise disjoint $\mathbb{Q}$-linearly independent sets of reals. Then $X = \bigcup_n X_n$ has zero inner measure.

Proof: Suppose not and let $K$ be a compact set of positive length contained in $X$. Let $A = \{(x, y) : y \in K + x\}$. By Lemma 1, there are disjoint perfect sets $P, Q$ such that $P \times Q \subseteq A$. Hence for every $x \in P, y \in Q$, $y - x \in K$. Consider the complete bipartite graph $K_{P, Q}$. Let $c$ be an edge coloring of $K_{P, Q}$ such that $c(x, y) = n$ iff $y - x \in X_n$. Let $P' \in [P]^{\omega_1}, Q' \in [Q]^{\omega_2}$. Using Fodor's lemma get $x_0, x_1 \in P'$, $y_0, y_1 \in Q'$ such that $c(x_i, y_j) = n^{\star}$ for every $i, j < 2$. Let $a = y_1 - x_0$, $b = y_1 - x_1$, $c = y_0 - x_1$, $d = y_0 - x_0$. Then, $a, b, c, d \in X_{n^{\star}}, \{a, c\} \cap \{b, d\} = \phi$ but $a-b+c-d = 0$: A contradiction.

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  • $\begingroup$ Fascinating answer. Why the change from $\aleph_3$ in Claim 2 to $\aleph_2$ in Claim 3? $\endgroup$ – Avshalom May 12 '15 at 10:15
  • $\begingroup$ Claim 3 is the Erods-Kakutani conjecture ($2^{\omega} \geq \omega_2$ is the negation of CH). Claim 2 is just an intermediate lemma in the proof of Claim 1. $\endgroup$ – Ashutosh May 12 '15 at 10:57
  • $\begingroup$ I follow. Claim 1/Lemma 1 wants $P$ and $Q$ disjoint. Is there a forcing-free proof of Claim 1? Many thanks for the lovely argument. $\endgroup$ – Avshalom May 12 '15 at 11:04
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    $\begingroup$ The reference for Mycielski's result is: J. Mycielski, Algebraic independence and measure, Fund. Math. 61 (1967), 165-169. I never read it but I don't think it uses forcing. $\endgroup$ – Ashutosh May 12 '15 at 11:19
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    $\begingroup$ "Kunen and I proved that if $c>\aleph_1$ then the union of $\aleph_0$ rationally independent sets always has inner measure 0." (then the proof, essentially the same as yours, is sketched) P. Erdos: Set theoretic, measure theoretic, combinatorial, and number theoretic problems concerning poit sets in Euclidean space, Real Analysis Exchange 4(1978-79), 113--138. $\endgroup$ – Péter Komjáth May 12 '15 at 15:46

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