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Statement

Here $f(n)$ is the function of the Theorem (p. 307) and $w=w(n)=(1/40)\sqrt{\log n(\log\log n)^{-1}}$ is the quantity of display (3.2) (p. 310), logarithms to base 2.

Remark (p. 313, unnumbered, closing the paper). The authors state that the lower-bound process of Section 3 can be iterated to give $f(n)>\sqrt n\,w^2/2$, and after $k$ iterations $f(n)>\sqrt n\,w^k/k!$, which in their words "grows up as high as $n^{1/2+\epsilon}$". They do not carry this out, citing messy details. The remark ends: "It is conceivable that $f(n)>n^{1-\epsilon}$ for every $\epsilon$ and $n\ge n_0(\epsilon)$."

The final sentence is an expectation, not a proved statement, and the iterated bounds are announced without proof. Since $\sum_kw^k/k!=e^w$ and $e^w=n^{o(1)}$ for the $w$ of (3.2), the announced bounds are all at most $\sqrt n\,n^{o(1)}$, so the $\epsilon$ in "as high as $n^{1/2+\epsilon}$" tends to $0$ with $n$ (an observation made here). The last sentence, for every $\epsilon>0$, is equivalent to $f(n)\ge n^{1-o(1)}$.

Source. S. L. G. Choi, J. Komlós and E. Szemerédi, On sum-free subsequences, Trans. Amer. Math. Soc. 212 (1975), 307--313, DOI 10.1090/S0002-9947-1975-0376594-1; the remark on printed p. 313, with (3.2) and footnote 2 on p. 310, read in the journal's printing. The paper and its edition are recorded on the source card.

Read depth. Claims checked: the remark was read clause by clause on the page image. It carries no proof.

Proof pointer

None: the paper gives no proof of the iterated bounds and none is claimed for the last sentence.

Dependencies

The lower-bound argument of the Theorem (Section 3, pp. 310--313), which the remark proposes to iterate.

Bears on

• Problem 790: the last sentence is what the site's commentary renders as the authors' "conjecture that $l(n)\geq n^{1-o(1)}$", the paper's $f(n)$ being the problem's $l(n)$. If true it would answer the problem's second displayed question, whether $l(n)<n^{1-c}$ for some $c>0$, in the negative. The paper proves nothing toward it beyond the Theorem's lower bound.