Problem 41

Erdős Problem #41: declared status 'open'. Formalized: yes. 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$$\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?$$ Current best: Erd\H{o}s proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then$$\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.$$This is discussed in problem C11 of Guy's collection [Gu04], in which Guy says Erd\H{o}s offered \$500 for the general problem of whether, for all$h\geq 2,\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/h}}=0\]whenever the sum of h$terms in$A$are distinct. Prize:$500. Tags: additive combinatorics, number theory, sidon sets.