we come to the main result of this work.

A theorem on the length of rainbow paths
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In this section, we are going to prove the following result, which is equivalent to Theorem \[thm:intro\].

\[thm:awesome\] For every $\epsilon > 0$, there is some $n_0 = n_0(\epsilon)$ such that for $n> n_0$, every proper edge coloring of $K_n$ contains a rainbow path on at least $(3/4-\epsilon)n$ vertices.

The proof proceeds by contradiction, that is, we assume that for some $\epsilon > 0$, there are no such rainbow paths, and show that we can construct a rainbow path that is longer than the length of a (supposedly) maximum rainbow path. There are two steps to how this is done:

1.  We show that there is a maximum rainbow path $P$ such that almost all (i.e., all but constantly many) vertices in $V(P)^c$ have at least some constant number of new neighbors in $V(P)^c$.

2.  We show how such a rainbow path can be extended to a longer rainbow path, resulting in a contradiction.

### Preliminaries

Let $\epsilon > 0$ and let $n>n_0$ for some suitably large value $n_0 = n_0(\epsilon)$. We will not give an explicit value for $n_0$, rather we 