Consider you want to properly edge color a graph such that it has no bi-colored cycle. Denote by $\alpha'(G)$ the least number of colors required to color the edges of $G$ in such a way.

It is well known that $\alpha'(G)$ is linear in $\Delta(G)$ and the best current bound seems to be $$\alpha'(G) \leq 4\Delta(G)-4,$$ obtained by what they call the entropy compression method.

Now consider you want to properly edge color your graph such that no 5-path is bi-colored and denote this number by $\beta'(G).$ Repeating the same argument with entropy compression one obtains an upper bound of order $O(\Delta(G)^{1.5})$ for $\beta'(G).$

If we consider the complete graph then it follows from here that $$\beta'(K_n) \leq n^{1+\epsilon}$$ for any fixed $\epsilon > 0.$

So while a linear number of color suffices to properly edge color a graph avoiding bi-colored cycles of **any** lenght it seems that the same problem is already much more complicated if we want to avoid just bi-colored 5-paths.

Given this I am wondering

Can you properly edge color $K_n$ such that no $5$-path is bi-colored and so that only a number of colors that is linear in $n$ is used?