Please have a look at the problem below.
Given a 3-uniform hypergraph $H=(V, E),$ the matching number $\nu(H)$ is the maximum number of pairwise-disjoint edges in $E(H) .$ The cover number $\tau(H)$ is the size of the smallest set of vertices $A \subseteq V(H)$ such that for every edge $e \in E(H)$ we have $e \cap A \neq \emptyset .$
A sail $S$ is a 3 -uniform hypergraph with four edges arranged as follows: there is a vertex $w \in V(S)$ contained in three of the edges, $e_1, e_2, e_3 \in E(S)$. These three edges contain unique vertices $x \in e_1 \setminus (e_2 \cup e_3), y \in e_2 \setminus (e_1 \cup e_3)$, and $z \in e_3 \setminus (e_1 \cup e_2)$. Finally, the fourth edge $e_4= \{x, y, z\}$ contains precisely these three vertices. Note that there are, up to isomorphism, three different types of sails, depending on which of the edges $e_1, e_2$ and $e_3$ share their third vertices.
Show that, for every $\varepsilon>0$ and $n$ sufficiently large, there is a $n$-vertex sail-free $3$-uniform hypergraph with $\tau(H) \geq(3-\varepsilon) \nu(H)$.
As a first step, I’m trying to characterize sail-free $3$-uniform hypergraph. I think either we forbid any three edges to share a vertex $w$, or, if they do, then we forbid any possible $e_4$, i.e., the edges formed by taking one vertex from each of $e_1,e_2,e_3$, apart from $w$. The resulting is the following picture (It’s better to visualize the vertex set as a $3$-d shape, with the forbidden edges represented by solid-line triangle. Also I only draw $2$ of the $8$ triangles, but hopefully the idea is clear).
Would this be a sensible condition to impose on the graph?