Problem 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\[ f(N)\sim N^{1/3}?\]
Categories:
Additive Combinatorics Sidon Sets
Progress
Originally asked to Erdős by Bose. Bose and Chowla [BoCh62] provided a construction proving one half of this, namely\[(1+o(1))N^{1/3}\leq f(N).\]The best upper bound known to date is due to Green [Gr01],\[f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}\](note that $(7/2)^{1/3}\approx 1.519$).More generally, Bose and Chowla conjectured that the maximum size of $A\subseteq \{1,\ldots,N\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then\[\lvert A\rvert \sim N^{1/r}.\]This is known only for $r=2$ (see [30]).
This is discussed in problem C11 of Guy's collection [Gu04].
Source: erdosproblems.com/241 | Last verified: January 14, 2026