Pin down $t(r)$: transversal number forced by a local $\tau\leq 1$ condition on $r$-uniform hypergraphs (Erdős #616)

Let $r\geq 3$ and let $G$ be an $r$-uniform hypergraph (every edge is a set of $r$ vertices). The transversal number (or covering number) $\tau(G)$ is the minimum size of a set of vertices that meets every edge. Suppose $G$ has the local property that every sub-hypergraph $G'$ induced on at most $3r-3$ vertices satisfies $\tau(G')\leq 1$ (a single vertex covers all edges of $G'$). Determine the best-possible constant $t=t(r)$ such that this local condition forces the global bound $\tau(G)\leq t$.

Acceptance. FULLY RESOLVES: determine $t(r)$ exactly — a closed form for all $r\geq 3$, or the exact limiting constant $\lim_{r\to\infty} t(r)/r$ together with a proof it exists — supported by a complete proof of both a construction (an $r$-uniform hypergraph in which every $\leq 3r-3$ vertices have $\tau\leq 1$ yet the global $\tau$ attains the claimed value) and a matching upper bound. ADVANCES: prove a new upper or lower bound on $t(r)$ that strictly improves the interval $[\tfrac{3}{16}r+\tfrac{7}{8},\ \tfrac{1}{5}r]$ stated in the background — e.g. a construction giving leading coefficient greater than $3/16$, or an argument giving leading coefficient below $1/5$ — with a full proof (for a construction, an explicit parametric hypergraph plus a verifier certifying both the local $\tau\leq 1$ property and the attained global $\tau$); or settle $t(r)$ exactly for a specific small $r$ by exhaustive computation with an optimality certificate. Deliver the proof, or the construction generator plus verification code and the attained constant.