ИСТИНА

Let k be a natural number. The property B_k-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. We say a hypergraph is simple if any two edges of it have no more than one vertex in the intersection. Define a quantity m∗_ k(n) which is equal to minimum possible number of edges in a simple n-uniform hypergraph that does not have property B_k. We obtain new lower and upper bounds for m∗_k(n).