  It is known a vector linear solution may exist if and only if the
characteristic of the finite field belongs to a certain set of primes. But, can
increasing the message dimension make a network vector linearly solvable over a
larger set of characteristics? To the best of our knowledge, there exists no
network in the literature which has a vector linear solution for some message
dimension if and only if the characteristic of the finite field belongs to a
set $P$, and for some other message dimension it has a vector linear solution
over some finite field whose characteristic does not belong to $P$. We have
found that by \textit{increasing} the message dimension just by $1$, the set of
characteristics over which a vector linear solution exists may get arbitrarily
larger. However, somewhat surprisingly, we have also found that by
\textit{decreasing} the message dimension just by $1$, the set of
characteristics over which a vector linear solution exists may get arbitrarily
larger.
  As a consequence of these finding, we prove two more results: (i) rings may
be superior to finite fields in terms of achieving a scalar linear solution
over a lesser sized alphabet, (ii) existences of $m_1$ and $m_2$ dimensional
vector linear solutions guarantees the existence of an $(m_1 +
m_2)$-dimensional vector linear solution only if the $m_1$ and $m_2$
dimensional vector linear solutions exist over the same finite field.
