  The antiferromagnetic $q$-state Potts model is perhaps the most canonical
model for which the uniqueness threshold on the tree is not yet understood,
largely because of the absence of monotonicities. Jonasson established the
uniqueness threshold in the zero-temperature case, which corresponds to the
$q$-colourings model. In the permissive case (where the temperature is
positive), the Potts model has an extra parameter $\beta\in(0,1)$, which makes
the task of analysing the uniqueness threshold even harder and much less is
known.
  In this paper, we focus on the case $q=3$ and give a detailed analysis of the
Potts model on the tree by refining Jonasson's approach. In particular, we
establish the uniqueness threshold on the $d$-ary tree for all values of $d\geq
2$. When $d\geq3$, we show that the 3-state antiferromagnetic Potts model has
uniqueness for all $\beta\geq 1-3/(d+1)$. The case $d=2$ is critical since it
relates to the 3-colourings model on the binary tree ($\beta=0$), which has
non-uniqueness. Nevertheless, we show that the Potts model has uniqueness for
all $\beta\in (0,1)$ on the binary tree. Both of these results are tight since
it is known that uniqueness does not hold in the complementary regime.
  Our proof technique gives for general $q>3$ an analytical condition for
proving uniqueness based on the two-step recursion on the tree, which we
conjecture to be sufficient to establish the uniqueness threshold for all
non-critical cases ($q\neq d+1$).
