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Claim. Theorem 1.2 of A. Beker, The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations, arXiv:2501.10203, p. 3: "Let $c\in(0,\frac1{68})$ be arbitrary. Then for any sufficiently large finite set $A\subseteq\mathbb Z$, there exists a subset $B\subseteq A$ of size at least $(\log|A|)^{1+c}$ such that $b_1+b_2\notin A$ for any distinct $b_1,b_2\in B$." In the notation of Problem 787,

$$g(n)\ge(\log n)^{1+c}\qquad\text{for every }c<\tfrac1{68}\text{ and all large }n,$$

which the paper states as $\phi(n)=\Omega((\log n)^{1+c})$ with $c=1/69$. It is deduced from Theorem 1.1 (for $\alpha\in(0,1]$ and $k\ge2$, a set of density $\alpha$ in $[N]$ with $N\ge\exp(Ck^{68}\log(2/\alpha)^{16})$, $C$ an absolute constant, contains a non-degenerate $k$-configuration), with Sanders's Proposition 2.7 in explicit form. The paper claims no improvement in the value of $c$ over Sanders's theorem and notes that the $k$-configuration route is limited to $c<1$; its gain is an explicit exponent, which the site renders as $(\log n)^{1+1/68+o(1)}$. Cited as [Be25] on the problem page. Library home beker_2025_erdos_moser_sum_free_set_problem; result page Theorem 1.2.

Covers. The lower bound $g(n)\ge(\log n)^{1+c}$ for every $c<1/68$, an explicit form of Sanders's bound. Not covered: the order of growth of $g(n)$.

Depends on. Sanders's theorem, whose deduction of the bound from Proposition 2.7 the paper follows, and Choi's 1971 paper for the reduction from real sets to integer sets (Beker's footnote 1).

Standing. Claimed. The arXiv v2 of 2 October 2026 (24 pages) is the final version, which its arXiv comment says incorporates the referee's comments, corrects an error in the proof of Lemma 2.2 of v1, and is to appear in International Mathematics Research Notices; Theorems 1.1 and 1.2 are unchanged. No journal record existed on 2026-10-07, so refereed is not listed until the publication appears. The site's curator, Thomas F. Bloom, credits the bound to Beker in the problem page's commentary (label OPEN, page last edited 23 January 2026); the problem is not marked settled there, so the credit is not reviewed. The statements are quoted from v1; neither proof is checked in this corpus.