Representations of $\mathfrak{su}(3)$ I am confused about the notation for representations of $\mathfrak{su}(3)$.  Often a bold number is used to denote a particular representation e.g. $\mathbf{3}$ is used to denote the fundamental representation and $\mathbf{8}$ denotes the adjoint representation.  Presumably, this notation is only justified if there is a unique representation of a given dimension, $n$, and then we can denote this representation by $\mathbf{n}$.  I don't know how to prove this if it is true.  If it is not true, then what is meant by the representation $\mathbf{n}$?

It is not true that a (finite-dimensional, complex) representation of $\mathfrak{su}(3)$ is determined by its dimension. In fact it's already the case, as James Cook points out in the comments, that the defining representation $V$ (which I think is the representation you denote by $\mathbf{3}$) and its dual $V^{\ast}$ are not isomorphic. One way to see this is that the corresponding representations of $\text{SU}(3)$ are not isomorphic because they have different characters. 
So I think this is a notational convention as you suspect in the comments. It is true that representations of $\mathfrak{su}(2)$ are determined by their dimension, so maybe that's where the convention comes from. Incidentally, the representation theory of all of these classical Lie algebras is completely understood; pop open any textbook on the subject to find the general statements. 
