Fix a positive integer constant $C$.

Let $S_1, S_2, \cdots , S_k$ be subsets of $\{ 1, 2, \cdots , N \}$. Let us call $S_1, S_2, \cdots , S_k$ a $C$-cover if for every subset $T$ of $\{ 1, 2, \cdots , N \}$, there exists $i_1, i_2, \cdots , i_C$ not necessarily distinct such that $T\subseteq S_{i_1} \cup S_{i_2} \cup \cdots \cup S_{i_C}$ and $C \cdot |T| > |S_{i_1} \cup S_{i_2} \cup \cdots \cup S_{i_C}|$

For all positive integers $N$, What is smallest such $k$ so that there exists $k$ subsets that form a $C$-cover.

I'm not looking for an exact answer but rather asymptotic bounds. Specifically, I'm wondering if the minimal $k$ is polynomial in $N$.

I'm sorry if I'm stating this really badly.