On Voloshin colorings in 3-hypergraph designs

Let P-(3) (2, 4) be the hypergraph having vertex set {1, 2, 3, 4} and edge set {{1, 2, 3}, {1, 2, 4}}. In this paper we consider vertex colorings of P-(3) (2, 4)-designs in such a way any block is neither monochromatic nor polychromatic. We find bounds for the upper and lower chromatic numbers, showing also that these bounds are sharp. Indeed, for any admissible v there exists a P-(3)(2, 4)-design of order v having the largest possible feasible set. Moreover, we study the existence of uncolorable P-(3)(2, 4)-designs, proving that they exist for any admissible order v >= 28, while for v <= 13 any P-(3) (2, 4)-design is colorable. Thus, a few cases remain open, precisely v = 14, 16, 17, 18, 20, 21, 22, 24, 25, 26.