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The extremal problem of hypergraph colorings related to Erd\H{o}s--Hajnalproperty $B$-problem is considered. Let $k$ be a natural number. The problem isto 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 thatevery edge of the hypergraph contains at least $k$ vertices of each color. Inthis paper we obtain new lower bounds for $m_k(n).$

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