  It is shown that in $d$-dimensional systems, the vertex corrections beyond
the random phase approximation (RPA) or GW approximation scales with the power
$d-\beta-\alpha$ of the Fermi momentum if the relation between Fermi energy and
Fermi momentum is $\epsilon_{\rm f}\sim p_{\rm f}^\beta$ and the interacting
potential possesses a momentum-power-law of $\sim p^{-\alpha}$. The condition
$d-\beta-\alpha<0$ specifies systems where RPA is exact in the high-density
limit. The one-dimensional structure factor is found to be the interaction-free
one in the high-density limit for contact interaction. A cancellation of RPA
and vertex corrections render this result valid up to second-order in contact
interaction. For finite-range potentials of cylindrical wires a large-scale
cancellation appears and found to be independent of the width parameter of the
wire. The proposed high-density expansion agrees with the Quantum Monte Carlo
simulations.
