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Statement

Conjecture (p. 460, unnumbered). Following the proof of Theorem 2, the paper says it seems likely that a stronger theorem also holds: if the continuum hypothesis is false and each of the sets $M_1,M_2,\ldots$ consists of rationally independent real numbers, then $\bigcup_{n=1}^{\infty}M_n$ has inner measure $0$. It is stronger than the converse half of Theorem 2, which says only that such a union is not the whole real line.

The paper gives no proof and no partial result toward it.

Read depth

Claims checked: the statement was read clause by clause on the page image of the print. Nothing here is independently reviewed.

Dependencies

None.

Source. P. Erdős and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457--461, doi:10.1090/S0002-9904-1943-07954-2; the edition read is named on the source card.

Bears on

No Erdős problem in the corpus.