  We generalise a key result of one-relator group theory, namely Magnus's
Freiheitssatz, to partially commutative groups, under sufficiently strong
conditions on the relator. The main theorem shows that under our conditions, on
an element $r$ of a partially commutative group $\mathbb{G}$, certain Magnus
subgroups embed in the quotient $G=\mathbb{G}/N(r)$; that if $r=s^n$ has root
$s$ in $\mathbb{G}$ then the order of $s$ in $G$ is $n$, and under slightly
stronger conditions that the word problem of $G$ is decidable. We also give
conditions under which the question of which Magnus subgroups of $\mathbb{G}$
embed in $G$ reduces to the same question in the minimal parabolic subgroup of
$\mathbb{G}$ containing $r$. In many cases this allows us to characterise
Magnus subgroups which embed in $G$, via a condition on $r$ and the commutation
graph of $\mathbb{G}$, and to find further examples of quotients $G$ where the
word and conjugacy problems are decidable. We give evidence that situations in
which our main theorem applies are not uncommon, by proving that for cycle
graphs with a chord $\Gamma$, almost all cyclically reduced elements of the
partially commutative group $\mathbb{G}(\Gamma)$ satisfy the conditions of the
theorem.
