= Solution
The kinetic term fixes the <mass dimension> of a canonically normalized <real scalar field> in $d$ dimensions to $(d-2)/2$. In six dimensions $[\phi]=2$, and every term in the <Lagrangian density> has dimension six. Therefore $[g_n]=6-2n$, and
$$
\boxed{n<3:\ \text{relevant};\qquad n=3:\ \text{marginal};\qquad n>3:\ \text{irrelevant}.}
$$
For positive integral $n$, these mean $n=1,2$, $n=3$ and $n\ge4$, respectively. If $n=0$ is included, its vacuum-energy constant is also relevant. These are classical <power counting in quantum field theory> classifications of <relevant couplings>, <marginal couplings> and <irrelevant couplings>, not assertions that loop corrections preserve exact marginality. For the remaining parts write $g=g_3$, assume $m^2>0$ and $0<\Lambda<\Lambda_0$, and define a local quartic coupling by $\int g_4\phi^4/4!$.
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