Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 241

claims/: The 1 claim page of Problem 241, one per claimant's result; the problem's standing derives from them.

Statement. Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences). Is it true that

f(N)\sim N^{1/3}?

Status. Open.

Source. erdosproblems.com/241, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #241, https://www.erdosproblems.com/241.

References.

[BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math. Helv. (1962/63), 141-147.
[Gr01] Green, Ben, The number of squares and $B_h[g]$ sets. Acta Arith. (2001), 365-390.
[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 Bose--Chowla lower bound for $A_h(n)$ , the upper bounds of Jia, Chen and Graham, and Helm's result that no sequence with $A(n)\sim\alpha n^{1/3}$ terms is a $B_3$ -sequence, with no value of $\alpha$ printed. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (10)

Bose and Chowla's lower bound for three-fold sums problems/additive_bases/E0241/claims Additive Bases and Sidon Sets Additive Bases and Sidon Sets 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/plagne_nd_recent_progress_finite_b_h_g Problem 6: does c_{h,g} = lim F_{h,g}(N)/N^{1/h} exist?Problem 9: estimate c_{3,1}, is c_{3,1} = 1 reasonable?number_theory/guy_2004_unsolved_problems_number_theory

Graph