Randomized algorithms for colourings of hypergraphs

Generalizations of the classical problem of Erdős on property $B$ of hypergraphs are studied. According to the definition of Erdős, a hypergraph has property $B$ if there exists a 2-colouring of its vertex set in which none of the edges of the hypergraph is monochromatic. It is required to find the quantity $m(n)$ that is equal to the minimal possible number of edges in an $n$-uniform (each edge contains exactly $n$ vertices) hypergraph that does not have property $B$. More general settings of the problem are considered, several parametric properties of hypergraphs are introduced, and the corresponding extremal quantities are studied. The main results improve the lower estimates of these extremal quantities that were known earlier.
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