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.