Problem 43 — Visual explainer (undergrad)
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Problem statement
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If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that\[ \binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1),\]where $f(N)$ is the maximum possible size of a Sidon set in $\{1,\ldots,N\}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound be improved to\[\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq (1-c+o(1))\binom{f(N)}{2}\]for some constant $c>0$?

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).

Notation (if it appears above)
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- `o(g(n))` means "grows strictly smaller than g(n)" (ratio → 0 as n→∞).
- `O(g(n))` means "at most a constant times g(n)" for large n.

Benchmarks / known results (from comments)
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- Since it is known that $f(N)\sim \sqrt{N}$ (see [30]) the latter question is
  equivalent to asking whether, if $\lvert A\rvert=\lvert B\rvert$,\[\lvert
  A\rvert \leq \left(\frac{1}{\sqrt{2}}-c+o(1)\right)\sqrt{N}\]for some
  constant $c>0$.
- In the comments Tao has given a proof of this upper bound without the $-c$.