Problem 41

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?

Source: erdosproblems.com/41

Number theory
Sidon sets
Additive combinatorics

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Status

Open

No claim settles this problem.

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.
Na89: Nash, John C. M., On $B_4$ -sequences. Canad. Math. Bull. 32 (4) (1989), 446-449.

Problem 41

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Proof claims

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