Hypergraphs with large transversal number

For k ≥ 2 , let H be a k -uniform hypergraph on n vertices and m edges. The transversal number τ ( H ) of H is the minimum number of vertices that intersect every edge. We consider the following question: Is τ ( H ) ≤ n / k + m / 6 ? For k ≥ 4 , we show that the inequality in the question does not always hold. However, the examples we present all satisfy Δ ( H ) ≥ 4 . A natural question is therefore whether τ ( H ) ≤ n / k + m / 6 holds when Δ ( H ) ≤ 3 . Although we do not know the answer, we prove that the bound holds when Δ ( H ) ≤ 2 , and for that case we characterize the hypergraphs for which equality holds. Furthermore, we prove that the bound holds when k = 2 (with no restriction on the maximum degree), and again there we characterize the hypergraphs for which equality holds. Chvátal and McDiarmid [V. Chvátal, C. McDiarmid, Small transversals in hypergraphs, Combinatorica 12 (1992) 19–26] proved that the bound holds for k = 3 (again with no restriction on the maximum degree). We characterize the extremal hypergraphs in this case.