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Nov 23, 2023 at 0:41 comment added Stanley Yao Xiao @fedja could you provide a reference or give a more precise statement? If so, I would gladly accept the answer, thank you!
Nov 22, 2023 at 23:34 comment added fedja This question (without a restriction on $N$) seems equivalent to a well-known problem about whether there is an infinite sequence of subsets $E_i$ of $[0,1]$ such that $|E_i|\ge \frac 12$ but $|E_i\cap E_j|<\frac 14$ for all $i\ne j$. Such a sequence does exist though you cannot replace $\frac 14$ by any smaller number here.
Nov 22, 2023 at 20:34 comment added Sam Hopkins You want $c$ to be independent of $N$?
Nov 22, 2023 at 20:25 history asked Stanley Yao Xiao CC BY-SA 4.0