A proof of Andersen's rainbow path conjecture for large $n$

Abstract

We show that, for sufficiently large $n$, every properly edge-coloured $n$-vertex complete graph contains a path with $n-1$ vertices which uses each colour at most once (that is, a rainbow path). This resolves a conjecture of Andersen from 1989 for all large $n$ and improves previous results of Alon-Pokrovskiy-Sudakov, and then Balogh-Molla, which showed that rainbow paths/cycles of length $n-O(n^{1/2}\log n)$ exist in this setting. Furthermore, with related methods, we show that, for every sufficiently large $n$, every Latin square of order $n$ contains a cycle-free transversal of order $n-2$, confirming a conjecture of Gy\'arf\'as and S\'ark\"ozy from 2014 for large $n$.