Transversals in Uniform Linear Hypergraphs

The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph has all edges of size $k$. It is known that $\tau(H) \le (n + m)/(k+1)$ holds for all $k$-uniform, linear hypergraphs $H$ when $k \in \{2,3\}$ or when $k \ge 4$ and the maximum degree of $H$ is at most two. It has been conjectured that $\tau(H) \le (n+m)/(k+1)$ holds for all $k$-uniform, linear hypergraphs $H$. We disprove the conjecture for large $k$, and show that the best possible constant $c_k$ in the bound $\tau(H) \le c_k (n+m)$ has order $\ln(k)/k$ for both linear (which we show in this paper) and non-linear hypergraphs. We show that for those $k$ where the conjecture holds, it is tight for a large number of densities if there exists an affine plane $AG(2,k)$ of order $k \ge 2$. We raise the problem to find the smallest value, $k_{\min}$, of $k$ for which the conjecture fails. We prove a general result, which when applied to a projective plane of order $331$ shows that $k_{\min} \le 166$. Even though the conjecture fails for large $k$, our main result is that it still holds for $k=4$, implying that $k_{\min} \ge 5$. The case $k=4$ is much more difficult than the cases $k \in \{2,3\}$, as the conjecture does not hold for general (non-linear) hypergraphs when $k=4$. Key to our proof is the completely new technique of the deficiency of a hypergraph introduced in this paper.