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Haviv and Levy, Symmetric complete sum-free sets in cyclic groups, Israel J. Math. 227 (2018), no. 2, 931--956, DOI 10.1007/s11856-018-1754-5 (Crossref record read); arXiv:1703.04118, first version 2017-03-12, the claim's date; an extended abstract appeared in Electron. Notes Discrete Math. 61 (2017), 585--591 (card).

The result. Theorem 1.5: there is a constant $c>0$ such that every sufficiently large cyclic group $\mathbb Z_n$ contains a symmetric complete sum-free subset of size at most $c\sqrt n$ (symmetric: $S=-S$; sum-free: no $a+b=c$ in $S$; complete: every element outside $S\cup\{0\}$ is a sum of two elements of $S$). The paper's Section 1 records the observation of Hanson and Seyffarth that the Cayley graph of $\mathbb Z_n$ with connection set $S$ is then an $|S|$-regular triangle-free graph of diameter $2$ on $n$ vertices: symmetry makes the graph undirected, sum-freeness excludes triangles, and completeness gives every nonadjacent pair a common neighbor. Hence $f(n)\le c\sqrt n$ for every large $n$, and with the trivial bound $f(n)\ge\sqrt{n-1}$ the order of growth of $f(n)$ is $\sqrt n$; the Erdős--Pach question whether $f(n)/\sqrt n\to\infty$ is answered no. The paper presents the theorem as extending Hanson and Seyffarth's construction, which covers the sequence $n=m^2+5m+2$, to every $n$. The constant $c$ is not made explicit, and the site's commentary records the paper as giving an alternative construction of the symmetric complete sum-free sets behind Hanson and Seyffarth's bound. The sequence case is Hanson and Seyffarth's and the sharper constant for all large $n$ is Füredi and Seress's; this claim uses neither.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication in the Israel Journal of Mathematics, cited with its venue above. The site's curator, Thomas Bloom, marks the problem DISPROVED and credits the paper, under the reference [HaLe18], with an alternative construction of the complete sum-free sets behind the bound; that credit is the reviewed evidence, and Bloom took no part in the paper. No independent review of the argument was made here; the theorem and the deduction to $f(n)$ were read, and no proof step of Theorem 1.5 was checked.