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Ok, I think I have figured it out. Define $I=\{\lambda\in [k] : \phi(N_\lambda) < \frac{1}{4k}\}$ and $J=[k]\setminus I$. Both the size of $I$ and $J$ is at most k. Define the event $\mathcal{E}_{(\lambda)}=\exists$ and index $s_\lambda$ such that $r_{s_\lambda}\in N_\lambda$. Clearly $\mathcal{E_s} = \cap_{\lambda\in[k]} \mathcal{E}_{(\lambda)}$. ...


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