Beker 2025 erdos moser sum free set problem

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proposition_4_1: The explicit form of Sanders's Proposition 2.7 from which Beker deduces Theorem 1.2: for k sufficiently large and sets of integers X inside Y with |X| >= exp(k^(68+o(1))) and |Y| <= (1 + k^-29)|X|, some k elements of X have no sum of two distinct ones in Y.

theorem_1_1: Beker's main bound for sets of integers lacking k-configurations: there is an absolute constant C > 0 such that for alpha in (0,1] and k >= 2, every subset of [N] of density at least alpha contains k distinct integers together with all their pairwise midpoints once N >= exp(C k^68 log(2/alpha)^16).

theorem_1_2: Beker's new proof, after Sanders's, of a lower bound of the form (log n)^(1+c) for the Erdős–Moser problem, deduced from his bound for sets lacking k-configurations: every sufficiently large finite set of integers A contains B of size at least (log |A|)^(1+c), for any fixed c below 1/68, with b_1 + b_2 outside A for distinct b_1, b_2 in B.

theorem_3_1: The finite-group form of Beker's k-configuration bound: in a finite abelian group G of odd order, a set A of density alpha > 0 contains all pairwise means of a uniformly random k-tuple with probability at least exp(-O(k^68 log(2/alpha)^16)), so |G| <= exp(O(k^68 log(2/alpha)^16)) when A contains no non-degenerate k-configuration.

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Adrian Beker, The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations. arXiv:2501.10203 (2025).

The copy read for this card is arXiv:2501.10203v1 (17 January 2025; 23 pages). The arXiv record lists a v2 of 2 October 2026 whose comment reads "Final version, incorporating the referee's comments and correcting an error in the proof of Lemma 2.2 in the previous version. To appear in International Mathematics Research Notices"; v2 was not read for this card, and labels and pages here are those of v1. Read status: claims checked for the definitions of $\phi(n)$ and of $k$-configurations and footnote 1 (pp. 1--2), Theorems 1.1 and 1.2 (p. 3) with the paragraph after Theorem 1.2, Theorem 3.1 (p. 8) with the deduction of Theorem 1.1 from it, and Section 4 (p. 17), each read clause by clause on the printed pages; the proofs of Theorem 2.1, Lemma 2.2 and Theorem 3.1 were not checked, and the paper omits the proof of Proposition 4.2. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2501.10203), every other right reserved.

A $k$-configuration, in an abelian group where doubling is an automorphism, is $k$ elements with the means of all their pairs, non-degenerate when the $k$ elements are distinct (p. 2). Theorem 1.1 (p. 3): for alpha in (0,1] and k >= 2, once N >= exp(C k^68 log(2/alpha)^16) with C an absolute constant, every subset of [N] of density at least alpha contains a non-degenerate k-configuration, improving Shao's bound exp(alpha^{-O(k^2)}) and extending the Kelley--Meka theorem (the case k = 2); taken separately, the dependence on log(2/alpha) is optimal up to its exponent by Behrend's construction, and the dependence on k by Green's random Cayley graph clique bound. Theorem 1.1 is deduced (p. 8) from Theorem 3.1, its form for finite abelian groups of odd order. Theorem 1.2 applies this to the Erdős--Moser problem: for any c in (0, 1/68) and any sufficiently large finite A in Z there is B in A with |B| >= (log |A|)^{1+c} and b_1 + b_2 not in A for distinct b_1, b_2 in B, so phi(n) = Omega((log n)^{1+c}) with c = 1/69, recovering the shape of Sanders's bound by a route through k-configurations, which Sanders bypassed. Section 4 obtains it from Proposition 4.1, an explicit form of Sanders's Proposition 2.7, proved by applying Theorem 3.1. The method combines the graph counting lemma of Filmus, Hatami, Hosseini and Kelman for binary systems of linear forms with Kelley--Meka techniques; the author must both handle forms depending on a single variable and make the density increment's dependence on k polynomial rather than exponential. The paper claims no improvement in the value of c over Sanders's and notes that the k-configuration approach is a priori limited to c < 1.

Source: https://arxiv.org/abs/2501.10203.

Bears on. #787: Theorem 1.2 (p. 3) gives, for each fixed $c\in(0,1/68)$, $g(n)\ge(\log n)^{1+c}$ for all large $n$ (through Choi's reduction from real sets to integer sets, footnote 1), a lower bound of the same shape as Sanders's with an explicit range for $c$. The site displays it as $(\log n)^{1+1/68+o(1)}\ll g(n)$; the theorem's range gives the exponent $1+1/68-o(1)$. Theorem 1.1, Theorem 3.1 and Proposition 4.1 are the steps from which the paper obtains it and state no bound for the problem themselves.

Results.

• Theorem 1.1 (p. 3): the bound for subsets of [N] lacking non-degenerate k-configurations.
• Theorem 1.2 (p. 3): the lower bound for the Erdős--Moser problem.
• Theorem 3.1 (p. 8): the form of Theorem 1.1 for finite abelian groups of odd order.
• Proposition 4.1 (p. 17): for k sufficiently large and X ⊆ Y ⊆ Z with |X| >= exp(k^{68+o(1)}) and |Y| <= (1 + k^{-29})|X|, some k elements of X are sum-free with respect to Y.

Theorem 2.1 (p. 5), the Section 2 graph counting lemma, a version of Theorem 2.1 of Filmus, Hatami, Hosseini and Kelman with the increment parameter delta = epsilon^2 m^{-2}/16000 for an oriented graph with m edges, and Lemma 2.2 (pp. 5--7), from which it is proved, are proof steps toward Theorem 3.1, summarized on its page; the v2 comment quoted above concerns the proof of Lemma 2.2.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.