Let $h = 7^q$ and $r = h + 1$. Expand the colored graph by a relation for every relation symbol with some positions filled by parameters from $U$, and by a unary relation $P_u$ for every $u ∈ U$. Formulas with parameters become formulas of the expansion without parameters and with the same quantifier rank. For a tuple far from $U$, the ball $B$ of radius $h$ around it in the expansion avoids $U$ and its neighbors, so the new relations are trivial on $B$, and the type of rank $q'$ of the tuple in the expansion of $B$ is determined by its type in the colored graph induced on $B$. By $LocalToGlobal$, that type determines the type of rank $q$ in the expansion, hence the type over $U$ in the colored graph. The types of rank $q'$ of $k$-tuples in colored graphs are bounded by the number of their finite codes, which depends on $q$, $k$ and $c$ only.