Problem: Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$.
Progress: Following are some of my thoughts on the problem:
Order the $c^5$ possible $5$-tuples of colors and let $t_k$ be the number of pairs $(i,j)$ with some simple $5$-edge-path from $v_i$ to $v_j$ having the $k$th tuple of colors. Note that for fixed $k,i,j$, there can be at most one such path from $v_i$ to $v_j$ (by the edge-incidence condition). Then we have $\sum_k t_k = \sum_{(i,j)} u_{i,j} = \Omega(n^6),$ and $\sum \binom{t_k}{2} \ge c^5\binom{\Omega(n^6)/c^5}{2} = c^{-5} \Omega(n^{12}).$ Now it suffices to show that $\sum \binom{t_k}{2} = O(n^4)$, but I don't know how to do this
Question: Prove that $\sum \binom{t_k}{2} = O(n^4)$.