Proof of Johnsen Lindenstrauss by concentration of measure

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Question: In Theorem 3 before "..By choosing $t=\sqrt{\frac{k}{5n}}$ we get $m\ge\Omega(\sqrt{\frac{k}{n}})$.." What is this process exactly? Why is an exponential bound critical here? Why wouldn't other type of bounds such as $\frac{1}{n}$ work?

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