On Perfect Balanced Rainbow-Free Colorings and Complete Colorings of Projective Spaces

This paper is motivated by the problem of determining the related chromatic numbers of some hypergraphs. A hypergraph Π _q(n,k) is defined from a projective space PG (n-1,q) , where the vertices are points and the hyperedges are (k-1) -dimensional subspaces. For the perfect balanced rainbow-free colorings, we show that χ_p(Π _q(n,k))=q^n-1/l(q-1) , where k≥⌈n+1/2⌉ and l is the smallest nontrivial factor of q^n-1/q-1 . For the complete colorings, we prove that there is no complete coloring for Π _q(n,k) with 2≤ k