Probability - Birthday paradox

Question

Let us assume every child in the world has a random and uniform favorite number between $1$ and $m$, and also has a different random and uniform unfavorite number between $1$ and $m$.
Denote $E_{k,m}$ the event in which, in a class with $k$ children there's a pair of children who chose exactly the same numbers (it doesn't matter favorite/unfavorite).
Prove that for any $\alpha>1$ it holds that:
$\lim _{m\rightarrow \infty }\left( \min _{k\in(m^{\alpha},\infty)} \mathbb{P}(E_{k,m})\right) = 1$.
I though about working with $E_{k,m}^{c}$ as done in the birthday paradox but I'm stuck on it and don't have any leads on how to show it