Problem 41 — Visual explainer (undergrad)
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Problem statement
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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\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?\]

Picture
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[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Benchmarks / known results (from comments)
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- 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, 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.
- This was proved for $h=4$ by Nash and for all even $h$ by Chen.