On hypergraph cliques with chromatic number 3 and a given number of vertices

In 1973, P. Erdos and L. Lovasz pointed out that any hypergraph with pairwise intersecting edges has chromatic number 2 or 3. In the first case, this hypergraph can have any number of edges. However, Erdos and Lovasz proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is n-uniform, has pairwise intersecting edges and chromatic number 3, the number of its edges is less than nn. Recently, D.D. Cherkashin improved this bound (see [2]). In this paper, we further improve it, when the number of vertices of an n-uniform hypergraph is bounded from above by the value nm with some m = m(n).