OPEN
This is open, and cannot be resolved with a finite computation.
- $100
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 $c>0$?
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 $-c$.
In the comments Barreto has given a negative answer to the second question: for infinitely many $N$ there exist Sidon sets $A,B\subset \{1,\ldots,N\}$ with $\lvert A\rvert=\lvert B\rvert$ and $(A-A)\cap (B-B)=\{0\}$ and\[\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\geq (1-o(1))\binom{f(N)}{2}.\]
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This page was last edited 20 December 2025.
Additional thanks to: Kevin Barreto and Terence Tao
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #43, https://www.erdosproblems.com/43, accessed 2026-02-13
