= Solution
The simplicial intersection conditions prohibit loops and multiple edges between the same pair of vertices, so the one-skeleton is a <simple graph> and
$$
E\leq\binom V2.
$$
Using the preceding identity gives $V^2-7V+6\chi\geq0$. A closed <surface> has $\chi\leq2$ and a simplicial <triangulation> has $V\geq4$. The smaller root is at most three, so this inequality forces $V$ to lie above the larger root:
$$
\boxed{V\geq\frac{7+\sqrt{49-24\chi}}2.}
$$
For completeness, fewer than four vertices could give at most one triangle under the intersection rules, which cannot form a closed <surface>.
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