On some generalizations of the property $B$-problem of an $n$-uniform hypergraph

The extremal problem of hypergraph colorings related to Erd\H{o}s–Hajnal property $B$-problem is considered. Let $k$ be a natural number. The problem is to find the value of $m_k(n)$ equal to the minimal number of edges in an $n$-uniform hypergraph not admitting $2$-colorings of the vertex set such that every edge of the hypergraph contains at least $k$ vertices of each color. In this paper we obtain new lower bounds for $m_k(n)\mathrm{.}$