The proof in the paper, with the padding step replaced by a split of each tuple into its entries outside $U$ and its entries inside $U$. Let $r = r(q)$ be the separation radius. For each $k' ≤ k$, the closure theorem for tuples gives polynomially many $s$-flips $H$ of $G$ such that every $k'$-tuple outside $U$ is at distance more than $r$ from $U$ in one of them. A flip $H$ is undone by a toggle on at most $s$ parts, so the edges of $G$ are quantifier-free definable in $H$ expanded by the colors and by predicates for the parts, and the type of a tuple in this expansion $H⁺$ determines its type in $G⁺$. A $k$-tuple $v̄$ is a rearrangement of ā ū with ā outside $U$ and ū inside $U$, and its type is determined by the type of ā ū and the positions. For fixed positions, ū, and $H$, the tuples ā far from $U$ in $H$ realize at most $T(q, k', c + s)$ types in $H⁺$. Summing over at most $2^k$ position sets, $(|U| + 1)^k$ tuples ū and the flips gives the bound, whose exponent does not depend on $c$.

The argument is carried out for the graph classes of the archive and their monadic dependence, and transferred to the graph classes used here.