On the balanced upper chromatic number of cyclic projective planes and projective spaces

We study vertex colorings of hypergraphs, such that all color sizes differ at most in one (balanced colorings) and each edge contains at least two vertices of the same color (rainbow-free colorings). Given a hypergraph H, the maximum k, such that there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number denoted by χ¯b(H). Concerning hypergraphs defined by projective spaces, bounds on the balanced upper chromatic number and constructions of rainbow-free colorings are given. For cyclic projective planes of order q we prove that: q2+q+16≤χ¯b(Πq)≤q2+q+13. We also give bounds for the balanced upper chromatic numbers of the hypergraphs arising from the n-dimensional finite space PG(n,q).