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The hypergraph chromatic number is the minimum number of colours in a proper hypergraph colouring. If $H$ has $N$ vertices and maximum hypergraph independent set size $a$, then $\chi (H)\ge N/a$, since the colour classes partition its vertex set into hypergraph independent sets.

A proper hypergraph colouring assigns colours to the vertices of a hypergraph so that no hypergraph edge is monochromatic. Each colour class is a hypergraph independent set. This is often called weak colouring, to distinguish it from requiring every pair of vertices in each edge to have different colours.

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.