= Solution
Let $L$ be the number of independent <loop momenta> in a connected <Feynman diagram>. Every <scalar propagator> falls as $k^{-2}$, and every <loop momentum> integration contributes three powers, so $\omega=3L-2e$. Counting line ends and using connectedness gives
$$
4n_4+6n_6=2e+n,\qquad L=e-n_4-n_6+1.
$$
Eliminating $L$ and $e$ yields
$$
\boxed{\omega=3(e-n_4-n_6+1)-2e=3-\frac n2-n_4.}
$$
Thus increasing the sextic <interaction vertex> count does not worsen the <superficial degree of divergence>; adding a quartic <interaction vertex> improves it. Only finitely many external-line structures can have an overall <ultraviolet divergence>, even though arbitrarily high <loop order> is possible.
After subtracting <ultraviolet subdivergences>, locality and <Lorentz invariance> restrict the divergent <counterterms> to a constant <vacuum energy density>, a <mass counterterm>, a <wave-function renormalization>, and quartic and sextic <field interaction terms>. Odd powers are forbidden by the $\phi\mapsto-\phi$ <Z2 symmetry>. In particular, $n=2$ permits a constant and $p^2$, $n=4$ permits only a constant, and $n=6$ permits only a constant. This is perturbative <renormalizability>, not a guarantee that every superficially allowed divergence actually occurs or that every unrenormalized graph with $\omega<0$ lacks divergent subgraphs.
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