keeping all other parameters fixed, for a given value of $Y_\nu$, the problem of vacuum stability becomes more acute with increasing number of right-handed neutrino species. This is expected since fermions contribute negatively to the evolution (see RG equations in Appendix \[app:seesaw\]), so extra right-handed neutrinos makes the vacuum more unstable. Fig. \[robustness\] also shows that a type-I seesaw extension will, at best, reach the [[Standard Model ]{}]{}curve (continuous line). Thus any (3,n) type-I seesaw extension of the [[Standard Model ]{}]{}with n $\geq$ 1 will:

1.  Necessarily be an inconsistent model for any $M^i_R \geq 10^{10}$ GeV; $i = 1, \cdots,n$

2.  For lower $M^i_R$ values the theory will still be inconsistent if the Higgs quartic self-coupling $\lambda$ goes negative before the largest mass scale $M^i_R$ in the theory [^4].

3.  In the best case scenario, i.e. for small enough Yukawas $Y^{aj}_\nu$ and masses $M^{ij}_R$, the type-I seesaw extensions can be self-consistent effective theories, but only in the regime where $\lambda$ is still positive. They should be embedded in a bigger theory.

Thus, in seesaw scenarios the stability properties of the electroweak vacumm will at best be those of the [[Standard Model ]{}]{}Higgs vacuum. In order to enhance Higgs vacuum 