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Statement

Question 8 (printed p. 35, quoted as printed). "Suppose $0<\alpha_1<\cdots<\alpha_k\le x$ is a sequence of real numbers with $k$ maximal such that any two sums $\sum_{j=1}^k\epsilon_j\alpha_j$, $\epsilon_j=0$ or $1$, differ by at least $1$. It is true [sic] that $k\le\frac{\log x}{\log2}+O(1)$? (This strengthens a well-known conjecture of Erdös.)"

The printed "It is true that" is read here as the question "Is it true that". The strengthening: a set $A\subseteq\{1,\ldots,N\}$ whose subset sums are distinct is the case of integers and $x=N$, since distinct integer sums differ by at least $1$, so a bound $k\le\log_2x+O(1)$ for reals gives $|A|\le\log_2N+O(1)$, that is $N\gg2^{|A|}$, the conjecture of Problem 1. The paper offers no result on the question.

Source. R. L. Graham, On sums of integers taken from a fixed sequence, Proceedings of the Washington State University Conference on Number Theory (1971), 22--40; Question 8 on printed p. 35 = PDF p. 14 of the author's publication-page scan, read on the page image (the scan has no text layer). The artifact is identified in the source digest.

Read depth. Claims checked: the question and its parenthetical were read clause by clause on the page image. A question; the paper proves nothing about it. Nothing here is independently reviewed.

Proof pointer

None. The paper asks the question and stops. The problem page records that the 2026 construction of Theorem 7.1 (for every $k$, a sum-distinct $A\subseteq\{1,\ldots,N\}$ with $kN<2^{|A|}$) also answers this real variant in the negative, its sets being integral; that standing and its qualifications are stated there, not here.

Dependencies

None.

Bears on

• Problem 1: the real variant recorded under the page's Known Results, in which $A\subset(0,N]$ and distinct subset sums must differ by at least $1$, is this question; the paper itself calls it a strengthening of Erdős's conjecture.