AI 中文总结
该研究证明了大n情形下Andersen的彩虹路径猜想，还证实了Gyárfás和Sárközy的相关猜想，改进了此前关于完全图中彩虹路径长度的结果。
AI 中文摘要
我们证明，对于足够大的n，每个边正常着色的n顶点完全图都包含一条有n-1个顶点的路径，该路径每种颜色最多使用一次（即彩虹路径）。这解决了Andersen在1989年提出的针对所有大n的猜想，并改进了Alon-Pokrovskiy-Sudakov以及Balogh-Molla之前的结果，此前这些结果表明该情形下存在长度为n-O(n^(1/2)log n)的彩虹路径/彩虹环。此外，利用相关方法，我们证明对于每个足够大的n，每个n阶拉丁方都包含一个n-2阶无环横截，证实了Gyárfás和Sárközy在2014年提出的针对大n的猜想。
英文摘要
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 Gyárfás and Sárközy from 2014 for large $n$.
Comments20 pages + 10 page appendix
