On upper transversals in 3-uniform hypergraphs

A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H. The upper transversal number Υ(H) of H is the maximum cardinality of a minimal transversal in H. We show that if H is a connected 3-uniform hypergraph of order n, then Υ(H) > (Formula presented). For n sufficiently large, we construct infinitely many connected 3-uniform(hypergraphs, H, of order n satisfying Υ(H) (Formula presented). We conjecture that sup (Formula presented), where n the infimum is taken over all connected 3-uniform ^n→∞ hypergraphs H of order n.