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Updated Oct 8

Problem 41

Statement. Let $A\subset\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that

\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?

Status. Open.

Source. erdosproblems.com/41, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #41, https://www.erdosproblems.com/41.

References.

[Ch96b] Chen, Sheng, A note on $B_{2k}$ sequences. J. Number Theory (1996), 1-3.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section C11 "Three-subsets with distinct sums", printed p. 184: the prize question $\liminf A_h(n)/n^{1/h}=0$ for infinite $B_h$ -sequences, settled for even $h$ and open for odd $h$ ; section E28 " $B_2$ -sequences. Mian-Chowla sequences.", printed p. 351 repeats the $h=3$ case as $\lim a_n/n^3=\infty$ . Library home: guy_2004_unsolved_problems_number_theory.
[Na89] Nash, John C. M., On $B_4$ -sequences. Canad. Math. Bull. 32 (4) (1989), 446-449.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (12)

Additive Bases and Sidon Sets Additive Bases and Sidon Sets additive_bases/fabian_2019_strong_infinite_sidon_b_h_sets Question 5.1 (p. 14): must every α-strong B_h set have lim inf S(n)/n^((1-α)/h) = 0?Theorem 1.2 (p. 3): α-strong B_h sets with exponent sqrt((h-1+α/2)^2+1-α) - (h-1+α/2)Theorem 1.3 (p. 3): every α-strong B_h set has S(n) <= c n^((1-α)/h)additive_bases/green_2001_number_squares_b_h_g_sets Theorem 17 (p. 15): B_3 sets in {1,...,N} have at most (7/2)^(1/3) N^(1/3)(1+o(1)) elements additive_bases/nash_1989_sequences Lemma 1: a block-count condition sum D_l^2 << N gives liminf C(n) (log n)^(1/2) / n^(1/2) finite Main theorem (3): a B_4-sequence has liminf A(n) (log n)^(1/4) / n^(1/4) finite number_theory/guy_2004_unsolved_problems_number_theory

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