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

Let $K_v^{\left(3\right)}=\left(X,ℰ\right)$ be the complete hypergraph, uniform of rank 3, defined on a vertex set $X=\left\{x_1,\text{…},x_v\right\}$, so that $ℰ$ is the set of all triples of $X$. Let $H^{\left(3\right)}=\left(V,D\right)$ be a subhypergraph of $K_v^{\left(3\right)}$, which means that $V\subseteq X$ and $D\subseteq ℰ$. We call 3-edges the triples of $V$ contained in the family $D$ and edges the pairs of $V$ contained in the 3-edges of $D$, that we denote by $\left[x,y\right]$. A $H^{\left(3\right)}$-design $\Sigma$ is called edge balanced if for any $x,y\in X$, $x\ne y$, the number of blocks of $\Sigma$ containing the edge $\left[x,y\right]$ is constant. In this paper, we consider the star hypergraph $S^{\left(3\right)}\left(2,m+2\right)$, which is a hypergraph with $m$ 3-edges such that all of them have an edge in common. We completely determine the spectrum of edge balanced $S^{\left(3\right)}\left(2,m+2\right)$-designs for any $m\ge 2$, that is, the set of the orders $v$ for which such a design exists. Then we consider the case $m=2$ and we denote the hypergraph $S^{\left(3\right)}\left(2,4\right)$ by $P^{\left(3\right)}\left(2,4\right)$. Starting from any edge-balanced $S^{\left(3\right)}\left(2,\frac{v+4}{3}\right)$, with $v\equiv 2\mathrm{mod}3$ sufficiently big, for any $p\in N$, $\left(\frac{v}{2}\right)\le p\le v$, we construct a $P^{\left(3\right)}\left(2,4\right)$-design of order $2v$ with feasible set $\left\{2,3\right\}\cup \left[p,v\right]$, in the context of proper vertex colorings such that no block is either monochromatic or polychromatic.