Problem 241 — Visual explainer (undergrad)
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Problem statement
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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\[ f(N)\sim N^{1/3}?\]

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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- Bose and Chowla 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,\[f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}\](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]).