An upper bound for the transversal numbers of 4-uniform hypergraphs

The main purpose of this paper is to prove that if H is a 4-uniform hypergraph with n vertices and m edges, then the transversal number r(H) <2(m +n)/9. All standard terminology of hypergraphs is from [ 11. Suppose H = (V, E) is a k-uniform hypergraph with n vertices and m edges. Tuza [2] proposed the problem of finding an upper bound for the transversal number r(H), of the form t(H) < ck(n + m), where ck depends only on k. More precisely, we want to determine ck z sup z(H)/(m + n), where H runs over all k-uniform hypergraphs of n vertices and m edges. It is easy to see that c1 = 4 and c2 = i. Tuza [2] proved that c3 = $ and asked if ck is 0(1/k). For any positive integer p we can construct a k-uniform hypergraph H