The Spectrum of P(5)(k,10-k)- designs

Abstract

Given an hypergraph H^(h), uniform of rank h, an H^(h)-design [or also a design of type} H^(h)] of order v is a pair Sigma=(X,B}), where X is a set of cardinality v and B is a collection of hypergraphs, all isomorphic to H^(h), such that every h-subset of X is an edge of exactly one hypergraph H^(h) belonging to B. An hyperpath P(h)/2 is an uniform hypergraph, having two non disjoint edges.
In this paper we determine the spectrum of hyperpath-designs of type P^(5)/2, iin the case that hyperedges have 3 or 4 vertices in common and formulate a conjecture about the cases k=1,2.