= Solution
Every <hypergraph independent set> in the induced <hypergraph> $H$ is also a <hypergraph independent set> in $H_0$. Thus its maximum independent-set size is at most $s-1<n/(2k)<N/k$. In any proper <hypergraph colouring> with at most $k$ colours, one <colour class> would have at least $N/k$ vertices, contradicting that bound. Therefore
$$
\boxed{\chi(H)>2005,\quad\text{hence}\quad\chi(H)\geq2006.}
$$
This is the weak <hypergraph colouring> convention: an <hypergraph edge> may use two colours, but it may not be <monochromatic>.
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