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Claim. S. L. G. Choi, On an extremal problem in number theory, J. Number Theory 6 (1974), no. 2, 105--111, printed p. 105: "Let $h(n)$ denote the largest function of $n$ such that from any set $\mathscr A$ of $n$ nonzero integers $a_1,\ldots,a_n$ one can always find a subset of $h(n)$ integers with the property that any two sums formed from its elements are equal only if they have equal number of summands. [...] The purpose of this paper is to obtain the estimate

$$h(n)\gg n^{1/3}(\log n)^{1/3}." \tag{1}$$

The sums run over subsets of the chosen set, so the summands are distinct, as in Problem 789. The paper's sets consist of nonzero integers; a set of $n$ integers containing $0$ has $n-1$ nonzero elements, so the bound holds for the site's $A\subseteq\mathbb Z$ with the same order, as the problem page explains. The proof (pp. 109--111) splits the set by the exact power of a prime $p\asymp(n\log n)^{1/3}$ dividing each element and, when no residue class is large and there are few classes, extracts from each large class of order $\log n$ integers with distinct subset sums modulo $p$ (the Lemma of p. 108). Cited as [Ch74b] on the problem page. Library home choi_1974_extremal_problem_number_theory; result page Estimate (1).

Covers. The lower bound $h(n)\gg(n\log n)^{1/3}$, which refines Erdős's $h(n)\ge n^{1/3}$ of 1965. Not covered: the order of growth of $h(n)$, which lies between this bound and Straus's $n^{1/2}$.

Depends on. No page of this wiki; the proof is self-contained apart from the Lemma it proves on p. 108.

Acceptance. Refereed: the paper is the publisher's version of record in the Journal of Number Theory (the Crossref record gives the issue of April 1974, with no day, so this page is named by the first of the month). The site's curator, Thomas F. Bloom, credits the improvement to $h(n)\gg(n\log n)^{1/3}$ to Erdős [Er62c] and Choi in the problem page's commentary (label OPEN); the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. The estimate is stated from printed p. 105; the proof is not reviewed in this corpus.