Zihong Tian’s research while affiliated with Hebei Normal University and other places

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 $\Pi _{q}(n,k)$.