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Claim. For each $1\le i\le4$ there is a constant $C_i$ such that for $n\equiv i\pmod4$

$$f_m(n)=(C_i+o(1))\,2^{n/4},$$

where $f_m(n)$ counts the maximal sum-free subsets of $\{1,\ldots,n\}$ as in Problem 877. This is the exact order of $f_m(n)$, so it settles the estimate the problem asks for and, a fortiori, answers the displayed question $f_m(n)=o(2^{n/2})$ yes. The source is Theorem 1.1 of J. Balogh, H. Liu, M. Sharifzadeh and A. Treglown, Sharp bound on the number of maximal sum-free subsets of integers, J. Eur. Math. Soc. 20 (2018), no. 8, 1885--1911, cited as [BLST18] on the problem page and cited from the arXiv version, with its result page on the library card. The paper says that each $C_i$ can be computed to any additive error in constant time and gives no closed form; its structural statement is that almost all maximal sum-free subsets of $\{1,\ldots,n\}$ look like one of two extremal constructions. The proof is unread. The result sharpens the same authors' exponent $1/4$ on its own page.

Depends on. No wiki page; the claim rests on the cited paper.

Acceptance. Refereed: the paper is a research article in the Journal of the European Mathematical Society (Crossref record read: volume 20, issue 8, published 4 June 2018); the text cited is arXiv:1502.07605v2 of 11 May 2018, whose comment says the paper is to appear there, and the journal text was not compared; the page is named by the first arXiv version, submitted 26 February 2015 (arXiv listing read). Reviewed: the site's curator, Thomas Bloom, marks the problem proved on erdosproblems.com and credits the four authors with this asymptotic, its constant determined by the residue of $n$ modulo $4$. Read status: the statement checked, the proof unread; no further evidence is listed.