On balanced colorings of hypergraphs

The paper deals with an extremal problem concerning hypergraph colorings. Let $k$ be an integer. The problem is to find the value $m_k(n)$ equal to the minimum number of edges in an $n$-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains $k$ vertices of each color. In this paper, we obtain the exact values of $m_2(5)$ and $m_2(4)$, and the upper bounds for $m_3(7)$ and $m_4(9)$.