Extremal problems for colourings of uniform hypergraphs

We study a classical problem (first posed by Erdős) in the extremal theory of hypergraphs. According to Erdős, a hypergraph possesses property $B$ if its set of vertices admits a 2-colouring such that no edge of the hypergraph is monochromatic. The problem is to find the minimum $m(n)$ of all $m$ such that there is an $n$-uniform (each edge contains exactly $n$ vertices) hypergraph with exactly $m$ edges that does not possess property $B$. We consider more general problems (including the case of polychromatic colourings) and introduce a number of parametric properties of hypergraphs. We obtain estimates for analogues of $m(n)$ for extremal problems on various classes of hypergraphs.