Ma and Tang: Largest Sidon subsets in weak Sidon sets

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theorem_1_2: States that the limit of g(n)/n exists and equals 1/2, where g(n) is the least possible size of a largest Sidon subset of an n-element weak Sidon set of reals, answering Problem 12 of Sárközy and Sós.

theorem_1_3: States that g(n), the least possible size of a largest Sidon subset of an n-element weak Sidon set of reals, equals the ceiling of (n+1)/2 for every positive integer n.

theorem_1_5: States that for (4,5)-sets the limit of f(n)/n exists and equals both the optimal constant c* of Erdős's problem and the infimum of f(n)/n over all positive integers n.

theorem_1_6: States that the optimal constant c*, such that every (4,5)-set of size n contains a Sidon subset of size at least c* n, satisfies 9/17 <= c* <= 4/7, improving both of the Gyárfás–Lehel bounds 1/2 + 1/(141·76) and 3/5.

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Jie Ma, Quanyu Tang, Largest Sidon subsets in weak Sidon sets. arXiv:2602.23282 (2026); the version read is v2 (6 March 2026). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2602.23282), every other right reserved.

Writing h(A) for the largest size of a Sidon subset of a finite set A of reals, the authors resolve Problem 12 of Sárközy and Sós by determining g(n), the minimum of h(A) over weak Sidon sets A of size n (sums of two distinct elements pairwise distinct): Theorem 1.3 (p. 2) gives g(n) = ceiling((n+1)/2) for every positive integer n, and Theorem 1.2 (p. 2) states that g(n)/n tends to 1/2, the existence of the limit coming separately from subadditivity of g (Section 3.1). For Erdős's problem on (4,5)-sets, in which any four distinct elements give at least 5 distinct values among their six pairwise absolute differences, Theorem 1.5 (p. 2) shows that the optimal constant c* equals lim f(n)/n = inf_{n>=1} f(n)/n, where f(n) is the analogous minimum over (4,5)-sets, and Theorem 1.6 (p. 2) improves the bounds 1/2 + 1/(14176) <= c <= 3/5 that the paper derives from Gyárfás and Lehel to 9/17 <= c* <= 4/7. The lower bound keeps the Gyárfás–Lehel reduction to transversal numbers of 3-uniform linear F_7-free hypergraphs and replaces their transversal estimate by one of Henning and Yeo; the upper bound comes from an explicit 14-point (4,5)-set whose largest Sidon subset has size 8, verified by exhaustive computer search (Lemma 5.1, p. 13), combined with the infimum characterization of c*. Every Sidon set is a (4,5)-set and every (4,5)-set is weak Sidon, both inclusions strict (Propositions 2.1 and 2.2, p. 4, and the examples on p. 5), so the weak Sidon result does not follow from the (4,5)-set work. The paper's Problem 1.4, the determination of c*, is the problem it identifies with Erdős Problem #757; the paper narrows both bounds and does not determine c*.

Source: https://arxiv.org/abs/2602.23282.

Results

Labels and pages are those of arXiv v2.

• Theorem 1.2 (p. 2): the limit of g(n)/n exists and equals 1/2, answering Problem 12 of Sárközy and Sós.
• Theorem 1.3 (p. 2): g(n) = ceiling((n+1)/2) for every positive integer n.
• Theorem 1.5 (p. 2): for (4,5)-sets the limit of f(n)/n exists and equals both c* and inf_{n>=1} f(n)/n.
• Theorem 1.6 (p. 2): 9/17 <= c* <= 4/7, with the upper bound resting on the computer-verified Lemma 5.1 (p. 13).

Read status. Claims checked for the four results above, read clause by clause on the print; the proofs were read for their structure, and the computer search of Lemma 5.1 was not rerun.

Bears on

• Problem 757: the problem's condition that every four-element subset B has |B-B| >= 11 is the (4,5)-set condition, so its constant is the paper's c*. Theorem 1.5 identifies c* with inf_{n>=1} f(n)/n, and Theorem 1.6 proves 9/17 <= c* <= 4/7; the paper does not determine c*.

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