Combinatorial Extremum Problems for $2$-Colorings of Hypergraphs

We consider a generalization of the Erdős–Hajnal classical combinatorial problem. Let $k$ be a positive integer. It is required to find the value of $m_k(n)$ equal to the minimum number of edges of an $n$-uniform hypergraph that does not admit $2$-colorings of the set of its vertices such that each edge of the hypergraph contains exactly $k$ vertices of each color. In the present paper, we obtain a new asymptotic lower bound for $m_k(n)$, which improves the preceding results in a wide range of values of the parameter $k$. We also consider some other generalizations of this problem.