rainbow path (a rainbow path visiting every vertex of $K_n$) (c.f. [@Maamoun1984]). Maamoun and Meyniel [@Maamoun1984] disproved this conjecture by constructing counterexamples for the case where $n$ is a power of two, as follows. Let $n=2^m$. Then we can identify the vertices of $K_n$ with distinct elements of the group $({\mathbold{Z}}/2{\mathbold{Z}})^m$, and color every edge $\{a,b\}$ of $K_n$ with the sum of the group elements corresponding to $a$ and $b$. This is a proper edge coloring, because for two edges $\{a,b\}$ and $\{a,c\}$, the group property implies $a+b\neq a+c$. Maamoun and Meyniel proved that this coloring admits no Hamiltonian rainbow paths. The reader is invited to check this fact for the case of $K_4$.

(-4,-2.5)(4,3.5) (0,0)[.08]{}[a]{} (0,3)[.08]{}[b]{} (-2.5891,-1.5,)[.08]{}[c]{} (2.5981,-1.5)[.08]{}[d]{}

Conversely, it is widely believed that in every proper edge coloring of $K_n$, there is a rainbow path on $n-1$ vertices (see for example [@GyarfasMhalla2010]). Still, this is far from proved, and to date, the best general lower bound on the number of vertices in a maximum rainbow path in (a properly edge colored) $K_n$ is $(2n+1)/3$, as proved by Gyárfás and Mhalla in [@GyarfasMhalla2010]. The main result of this thesis improves this bound to $(3/4-o(1))n$.

\[thm:intro\] In every proper edge 