Transversals and blocking sets in H(3)-designs

Abstract

If H^(h) is a subhypergraph of order  n of  K_v^(h), the complete and h-uniform hypergraph of order v, an H^(h)-decomposition of K_v^(h), also called an H^(h)-design of order v, is a pair Σ=(X,B), where B  is a partition of the  edge-set of K_v^(h) into classes generating hypergraphs all isomorphic to H^(h). The classes of the partition B are said to be the blocks of Σ. Using hypergraph terminology, if Σ=(X,B) is an H^(h)-design, a transversal T  of Σ is a subset of X intersecting every block of Σ. The transversal number  of Σ is the minimum number τ(Σ)=τ for which there exists a transversal of Σ having cardinality τ.  A blocking set B  of Σ is a subset of X such that both B  and C_X(B)  are transversals. In this paper, the existence of transversals and blocking sets in H⁽³⁾-designs are studied.