Bounds on the game transversal number in hypergraphs

Let H=(V,E) be a hypergraph with vertex set V and edge set E of order n_H=|V| and size m_H=|E|. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H. A vertex hits an edge if it belongs to that edge. The transversal game played on H involves of two players, Edge-hitter and Staller, who take turns choosing a vertex from H. Each vertex chosen must hit at least one edge not hit by the vertices previously chosen. The game ends when the set of vertices chosen becomes a transversal in H. Edge-hitter wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game transversal number, τ_g(H), of H is the number of vertices chosen when Edge-hitter starts the game and both players play optimally. We compare the game transversal number of a hypergraph with its transversal number, and also present an important fact concerning the monotonicity of τ_g, that we call the Transversal Continuation Principle. It is known that if H is a hypergraph with all edges of size at least  2, and H is not a 4-cycle, then τ_g(H)≤[formula parsented](n_H+m_H); and if H is a (loopless) graph, then τ_g(H)≤[formula parsented](n_H+m_H+1). We prove that if H is a 3-uniform hypergraph, then τ_g(H)≤[formula parsented](n_H+m_H), and if H is 4-uniform, then τ_g(H)≤[formula parsented](n_H+m_H).