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$\begingroup$ Sorry but this does not help... I have thought about this before, trying to fill in the central space. However, you could check the $2/3n$th entry on the first row, which is pointed by $2/3n$ (and no more) entries. This question seeks for maximin, not the average... $\endgroup$
– JetfiRex
Commented Jan 22, 2022 at 16:48
$\begingroup$ Oh I see. I guess it is all the $O(1)$ which gave me the idea of combining "asymptotic" and "average". Where you talk about average, you are rather applying a pigeonhole principle. $\endgroup$
– Wolfgang
Commented Jan 22, 2022 at 17:22

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