= Solution
Use upper indices for the defining <representation> of <SU(3)> and lower indices for its <dual representation>. A <Young diagram> with row lengths $(\ell_1,\ell_2)$ has <highest weight> <Dynkin labels> $(p,q)=(\ell_1-\ell_2,\ell_2)$; a full column of height three is a <determinant representation> and can be removed for <SU(3)>. The <Weyl dimension formula> reduces to
$$
\dim(p,q)=\frac{(p+1)(q+1)(p+q+2)}{2}.
$$
The six <irreducible representations> of smallest <dimensions> are $\mathbf1,\mathbf3,\overline{\mathbf3},\mathbf6,\overline{\mathbf6},\mathbf8$. The following box arrays draw their <Young diagrams>; filling boxes by tensor-slot labels gives representative <Young tableaux>:
$$
\begin{array}{c|c|c|c|c}
\text{representation}&\text{diagram}&(\ell_1,\ell_2)&(p,q)&\dim\\\hline
\mathbf1&\varnothing&(0,0)&(0,0)&1\\[2pt]
\mathbf3&\boxed{\phantom{i}}&(1,0)&(1,0)&3\\[2pt]
\overline{\mathbf3}&\begin{array}{|c|}\hline\phantom{i}\\\hline\phantom{i}\\\hline\end{array}&(1,1)&(0,1)&3\\[4pt]
\mathbf6&\begin{array}{|c|c|}\hline\phantom{i}&\phantom{i}\\\hline\end{array}&(2,0)&(2,0)&6\\[4pt]
\overline{\mathbf6}&\begin{array}{|c|c|}\hline\phantom{i}&\phantom{i}\\\hline\phantom{i}&\phantom{i}\\\hline\end{array}&(2,2)&(0,2)&6\\[4pt]
\mathbf8&\begin{array}{l}\boxed{\phantom{i}}\boxed{\phantom{i}}\\[-3pt]\boxed{\phantom{i}}\end{array}&(2,1)&(1,1)&8
\end{array}
$$
Their associated <tensors>, in the same order, can be taken as follows:
* $\mathbf1$: a scalar $s$, unchanged by <SU(3)>.
* $\mathbf3$: a <vector> $v^i$, with $v^i\mapsto U^i{}_jv^j$.
* $\overline{\mathbf3}$: an <antisymmetric tensor> $A^{[ij]}$, equivalently a <covector> $w_k=\tfrac12\epsilon_{kij}A^{ij}$, with $w_i\mapsto w_j(U^{-1})^j{}_i$.
* $\mathbf6$: a <symmetric tensor> $S^{(ij)}$, with $S^{ij}\mapsto U^i{}_kU^j{}_\ell S^{k\ell}$.
* $\overline{\mathbf6}$: a symmetric <covariant tensor> $S_{(ij)}$, with $S_{ij}\mapsto S_{k\ell}(U^{-1})^k{}_i(U^{-1})^\ell{}_j$. In the four-upper-index <Young tableau> description, use two antisymmetric index pairs with symmetry under exchange of pairs, and convert each pair to a lower index with the invariant $\epsilon$ <tensor>.
* $\mathbf8$: a mixed <tensor> $T^i{}_j$ with $T^i{}_i=0$, transforming as $T\mapsto UTU^{-1}$. This is the <Adjoint representation>. Its nine components obey one independent <trace> constraint.
These <low-dimensional SU3 tensor representations> have <dimensions> $1,3,3,6,6,8$ by direct component counting. To check that none were missed, if $p+q\geq3$ then $(p+1)(q+1)\geq p+q+1$, so the displayed <dimension> is at least $(p+q+1)(p+q+2)/2\geq10$. The pairs with $p+q\leq2$ are precisely the six listed pairs.
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