Problem 241

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}?

Source: erdosproblems.com/241

Additive combinatorics
Sidon sets

Wiki page

Status

Open

No claim settles this problem.

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.

Problem 241

Status

Proof claims

Discussion