$(\sigma, \tau, k+1)$-poor position $v$. Now let $a$ be a $\sigma$-labeled position. If $a \leq v$ then all $(d, \gamma, \tau)$-witnesses $b \in W(a)$ are contained in $Z$ by construction (as $a$ is one of the $k+1$ leftmost $\sigma$-labeled positions). If $a > v$ then all $(d, \gamma, \tau)$-witnesses of $a$ are among the last $k+1$ $\tau$-labeled positions which are also contained in $Z$. The argument for $\tau$-labeled positions is symmetric.

The structure ${\ensuremath{\mathfrak{A}}}'$ thus constructed satisfies (1) and (2).

It remains to construct ${\ensuremath{\mathfrak{B}}}$. If $|{\ensuremath{\mathfrak{A}}}'| > N$ for $N \geq c (|\Sigma||\Gamma|)^4$ for an appropriate constant $c$ then there are positions with successors $b_1$ and $b_2$ such that (i) there is no position $z \in Z$ with $a_1 < z < a_2$, (ii) $a_1$ and $a_2$ have the same unary $T$-type, and (iii) $(a_1, b_1)$ and $(a_2, b_2)$ have the same binary $T$-type. The solution ${\ensuremath{\mathfrak{B}}}$ is obtained from ${\ensuremath{\mathfrak{A}}}'$ by removing all elements between $a_1$ and $a_2$ (including $a_2$), and assigning to $(a_1, b_2)$ the binary type of $(a_1, b_1)$ in ${\ensuremath{\mathfrak{A}}}'$. Then ${\ensuremath{\mathfrak{B}}}$ satisfies all universal constraints of $C$ since no new types have been created. Further ${\ensuremath{\mathfrak{B}}}$ satisfies all existential constraints of $C$ since elements 