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Source. The unnumbered result on pp. 133--135 of P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79), no. 2, 113--138, doi:10.2307/44151159, as identified on the source card. Pages are those of the journal print.

Read depth. Claims checked: the statement (p. 133) was read clause by clause, the proof (pp. 133--135) for its structure. Nothing here is independently reviewed.

Statement

Assume $\mathfrak c>\aleph_1$. For $n=1,2,\ldots$ let $S_n$ be a set of real numbers in which all sums $x+y$ with $x,y\in S_n$ are distinct, which the paper glosses as all distances between points of $S_n$ being distinct. Then there are $\aleph_1$ rationally independent reals $b_\alpha$, $1\le\alpha<\omega_1$, and a real $t$ such that every number $t+\sum_\beta r_\beta b_\beta$, with rational $r_\beta$ and finitely many terms, lies outside $\bigcup_{n=1}^\infty S_n$ (p. 133). So the complement of the union contains a translate of an $\aleph_1$-dimensional linear subspace of the reals over the rationals.

The paper asks (p. 135) whether the translate can be dropped, the complement containing the rational span itself, and whether $\aleph_1$ can be replaced by $\aleph_2$, saying it does not think the latter likely. It adds that Baumgartner proved, answering an earlier question of Erdős, that the complement of a single set of reals with all sums $x+y$ distinct contains an infinite arithmetic progression, and that the proof here borrows from Baumgartner's unpublished proof (p. 135).

Proof pointer

Pp. 133--135. The argument of the Hajnal-Erdős lemma of Theorem 2 gives a set $A$ of $\aleph_2$ reals such that $ra$ lies outside the union for every rational $r$ and every $a\in A$. Against a fixed set $B$ of $\aleph_1$ rationally independent reals, call $a\in A$ bad when $\aleph_1$ numbers $ra+\sum c_\beta b_\beta$ lie in the union. Counting choices shows at most $\aleph_1$ elements of $A$ are bad, since $\aleph_2$ bad ones would put four numbers $u+v$, $u+v'$, $u'+v$, $u'+v'$ into one $S_n$, against distinct sums. A good $a$ then serves after discarding countably many elements of $B$. The printed proof writes the overlined union, the complement, at places where the union itself is meant (pp. 133--135).

Dependencies

The lemma of Hajnal and Erdős recorded on the Theorem 2 page.

Bears on

No Erdős problem page of the corpus cites this result.