Transversals in 4-Uniform Hypergraphs

Let H be a 4-uniform hypergraph on n vertices. The transversal number τ(H) of H is the minimum number of vertices that intersect every edge. The result in [J. Combin. Theory Ser. B 50 (1990), 129–133] by Lai and Chang implies that τ(H) 6 7n/18 when H is 3-regular. The main result in [Combinatorica 27 (2007), 473–487] by Thomassé and Yeo implies an improved bound of τ(H) 6 8n/21. We provide a further improvement and prove that τ(H) 6 3n/8, which is best possible due to a hypergraph of order eight. More generally, we show that if H is a 4uniform hypergraph on n vertices and m edges with maximum degree ∆(H) 6 3, then τ(H) 6 n/4 +m/6, which proves a known conjecture. We show that an easy corollary of our main result is that if H is a 4-uniform hypergraph with n vertices and n edges, then τ(H) 6 3 7n, which was the main result of the Thomassé-Yeo paper [Combinatorica 27 (2007), 473–487].