Polychromatic hypergraph colouring

A colouring is polychromatic with $r$ colours if every hypergraph edge contains a vertex of every colour. This is stronger than requiring every edge to be nonmonochromatic. Under independent uniform colours, a $k$-element edge misses at least one of three colours with probability $3(2/3{)}^k-3(1/3{)}^k$. If each edge meets at most $m$ other edges, the edge-intersection dependency graph of events and the Lovász local lemma therefore give a polychromatic three-colouring whenever $e(m+1)[3(2/3{)}^k-3(1/3{)}^k]\le 1$.

Take every three-element subset of a four-element vertex set as a hypergraph edge. Any three-colouring repeats a colour on two vertices. A triple containing that pair cannot contain all three colours. Thus this uniform hypergraph has no polychromatic three-colouring, although each edge meets only three other edges.