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 Upsilon(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 Upsilon(H) > 1.4855 3 root n - 2. For n sufficiently large, we construct infinitely many connected 3-uniform hypergraphs, H, of order n satisfying Upsilon(H) < 2.5199 3 root n. We conjecture that sup(n ->infinity) (inf Upsilon(H)/3 root n) = 3 root 16, where -F2, the infimum is taken over all connected 3-uniform hypergraphs H of order n.