 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. ThereforeThis is the weak hypergraph colouring convention: an hypergraph edge may use two colours, but it may not be monochromatic.