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Following the recent paper which initiated the study of colour isomorphismproblems for complete graphs, we obtain upper bounds for $f_2(n,H)$ for afamily of graphs $H$ obtained as the $K_0$-th rooted power of a balanced rootedtree for some sufficiently large $K_0$. The proof uses the random polynomialmethod of Bukh. We also obtain matching lower bounds for $1$-subdivisions ofthe complete bipartite graph.

On Color Isomorphic Pairs in Proper Edge Colourings of Complete Graphs