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 \(\Pi _{q}(n,k)\) 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 \({\overline{\chi }}_{p}(\Pi _{q}(n,k))=\frac{q^n-1}{l(q-1)}\), where \(k\ge \lceil \frac{n+1}{2}\rceil \) and l is the smallest nontrivial factor of \(\frac{q^n-1}{q-1}\). For the complete colorings, we prove that there is no complete coloring for \(\Pi _{q}(n,k)\) with \(2\le k<n\). We also provide some results on the related chromatic numbers of subhypergraphs of \(\Pi _{q}(n,k)\).