On algorithmic methods of analysis of two-colorings of hypergraphs

This 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 at least $k$ vertices of each color. In this paper, we obtain upper bounds of $m_k(n)$ for small $k$ and $n$, the exact value of $m_4(8)$, and a lower bound for $m_3(7)$.