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Claim. Every sequence of $n$ distinct integers has a sum-free subsequence, one in which no term is the sum of distinct other terms, of at least $c\,(n\log n/\log\log n)^{1/2}$ terms, and some sequence of $n$ distinct integers has no sum-free subsequence of more than $Cn/\log n$ terms: in the notation of Problem 790,

$$\Bigl(\frac{n\log n}{\log\log n}\Bigr)^{1/2}\ll l(n)\ll\frac{n}{\log n}$$

(the Theorem, display (1.1), printed p. 307). S. L. G. Choi, J. Komlós and E. Szemerédi, On sum-free subsequences, Trans. Amer. Math. Soc. 212 (1975), 307--313, cited as [CKS75] on the problem page. Library home choi_1975_sum_free_subsequences; result page Theorem. The paper's $f(n)$ is the site's $l(n)$, and its sum-free condition is the problem's. The lower bound answers the first displayed question: $l(n)n^{-1/2}$ tends to infinity. The upper bound shows that no subset of linear size is guaranteed, which also answers item 1.22 b) of the 1999 booklet in the negative, as Choi's paper in Proc. Amer. Math. Soc. 41 (December 1973) had already done with $l(n)\ll n(\log\log n)^{-1/2}$. The upper bound is witnessed by the explicit set $A_0\cup\dots\cup A_{s+1}$ with $A_i=2^i[t,2t)$ for $i\le s$, $A_{s+1}$ any $n-t(s+1)$ further integers, and $t=[n(\log n/3)^{-1}]$, through a lemma on sequences with few distinct pairwise sums; the lower bound extracts subsequences with monotone gaps and proves a proposition by induction over blocks. The paper's closing remark expects that iterating the lower-bound argument gives $n^{1/2+\epsilon}$ and finds it conceivable that $l(n)>n^{1-\epsilon}$ for every $\epsilon>0$ and all large $n$.

Covers. The first displayed question, answered yes, and the two bounds. Not covered: the estimate of $l(n)$ between the bounds and the second displayed question, whether $l(n)<n^{1-c}$ for some $c>0$ and all large $n$; the pending claim of 2026 asserts $l(n)\gg n/(\log n)^2$, which would answer it in the negative.

Depends on. No page of this wiki. The upper bound's proof is self-contained; the lower bound uses a theorem of Chvátal and Komlós on monotone consecutive differences (the paper's reference [5], Theorem 4; Canad. Math. Bull. 14 (1971), 151--157).

Acceptance. Refereed: the paper is the publisher's version of record in Transactions of the American Mathematical Society (as its Crossref record gives it; the volume carries no publication day, so this page is named by the first day of its year). The site's curator, Thomas F. Bloom, credits both bounds to Choi, Komlós and Szemerédi in the problem page's commentary (label OPEN, page last edited 23 January 2026), after a thread comment of 30 October 2025 pointed to the paper; the problem is not marked settled there, so the credit is recorded here and is not listed as reviewed. The Theorem and the closing remark are stated from the printed pages; the proofs are not reviewed in this corpus.