  We consider the initial value problem associated to a system consisting
modified Korteweg-de Vries type equations $$
  \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,\ \ v(x,0)=\phi(x),
  $$ $$ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) =0,\ \
w(x,0)=\psi(x),$$ and prove the local well-posedness results for given data in
low regularity Sobolev spaces $H^{s}(\textrm{I}\!\textrm{R})\times
H^{k}(\textrm{I}\!\textrm{R})$, $s,k> -\frac12$ and $|s-k|\leq 1/2$, for
$\alpha\neq 0,1$. Also, we prove that: (I) the solution mapping that takes
initial data to the solution fails to be $C^3$ at the origin, when $s<-1/2$ or
$k<-1/2$ or $|s-k|>2$; (II) the trilinear estimates used in the proof of the
local well-posedness theorem fail to hold when (a) $s-2k>1$ or $k<-1/2$ (b)
$k-2s>1$ or $s<-1/2$; (c) $s=k=-1/2 $; (III) the local well-posedness result is
sharp in a sense that we can not reduce the proof of the trilinear estimates,
proving some related bilinear estimates (as in Tao [19]).
