Edge balanced star-hypergraph designs and vertex colorings of path designs

Let (Formula presented.) be the complete hypergraph, uniform of rank 3, defined on a vertex set (Formula presented.), so that (Formula presented.) is the set of all triples of (Formula presented.). Let (Formula presented.) be a subhypergraph of (Formula presented.), which means that (Formula presented.) and (Formula presented.). We call 3-edges the triples of (Formula presented.) contained in the family (Formula presented.) and edges the pairs of (Formula presented.) contained in the 3-edges of (Formula presented.), that we denote by (Formula presented.). A (Formula presented.) -design (Formula presented.) is called edge balanced if for any (Formula presented.), (Formula presented.), the number of blocks of (Formula presented.) containing the edge (Formula presented.) is constant. In this paper, we consider the star hypergraph (Formula presented.), which is a hypergraph with (Formula presented.) 3-edges such that all of them have an edge in common. We completely determine the spectrum of edge balanced (Formula presented.) -designs for any (Formula presented.), that is, the set of the orders (Formula presented.) for which such a design exists. Then we consider the case (Formula presented.) and we denote the hypergraph (Formula presented.) by (Formula presented.). Starting from any edge-balanced (Formula presented.), with (Formula presented.) sufficiently big, for any (Formula presented.), (Formula presented.), we construct a (Formula presented.) -design of order (Formula presented.) with feasible set (Formula presented.), in the context of proper vertex colorings such that no block is either monochromatic or polychromatic.