This question is kind of weird, and it raised from an even weirder question (in the bins and balls model), but I've tried to simplify it as much as I can
let $n,k\in \mathbb N$ be 2 numbers such that $\log n\le k\le\sqrt n$, our job is to choose a number $x\in \mathbb N$
such that if $X\sim \text{Poisson}(x)$ we have
$\binom {2n}{2x}\Pr [X\ge k]\ge 1$
But! we want $x$ to be as low as possible
for $x=2k$ we have (by Markov's inequality) that probability is at least $\frac{1}{2}$ and the Binom coefficient is some positive integer $\ge 2$, so the inequality is satisfied, but I hoped that one could achieve something better, more specifically $x$ that is asymptotically less than $k$
It seems one cannot use the Chernoff bounds here, because we want to lower bound the probability of $X$ to be more than its expectancy