The Spectrum of Balanced P^(3)(1, 5)-Designs

Given a 3-uniform hypergraph H(3), an H(3)-decomposition of the complete hypergraph K(3)_v is a collection of hypergraphs, all isomorphic to H(3), whose edge sets partition the edge set of K(3)_v. An H(3)-decomposition of K(3)_v is also called an H(3)-design and the hypergraphs of the partition are said to be the blocks. An H(3)-design is said to be balanced if the number of blocks containing any given vertex of K(3)_v is a constant. In this paper, we determine completely, without exceptions, the spectrum of balanced P(3)(1 5)-designs

ON THE SPECTRUM OF OCTAGON QUADRANGLE SYSTEMS OF ANY INDEX

An \emph{octagon quadrangle} is the graph consisting of a length $8$ cycle $(x_{1},x_{2},\dots,x_{8})$ and two chords, $\{x_{1},x_{4}\}$ and $\{x_{5},x_{8}\}$. An \emph{octagon quadrangle system} of order $v$ and index $\lambda$ is a pair $(X,\mathcal B)$, where $X$ is a finite set of $v$ vertices and $\mathcal B$ is a collection of octagon quadrangles (called blocks) which partition the edge set of $\lambda K_{v}$, with $X$ as vertex set. In this paper we determine completely the spectrum of octagon quadrangle systems for any index $\lambda$, with the only possible exception of $v=20$ for $\lambda=1$