On two-colorings of hypergraphs

A study was conducted to address a problem arising in the context of the classical problem of P. Erdös and A. Hajnal in the extremal hypergraph theory. P. Erdös and A. Hajnal revealed that when an n-uniform hypergraph had sufficiently few edges then it had a natural analogue of bipartiteness. It was also revealed that when the hypergraph had many edges then it did not possess property B. An important generalization of property B was property Bk, which was proposed by A.M. Raigorodskii in 2003 and was originally studied by Shabanov. A hypergraph H = (V, E) was said to have property Bk when there existed a red-blue coloring of V such that each edge e ∈ E had at least k red vertices and at least k blue vertices.