论文标题
三角人家庭的彩虹三角
Rainbow triangles in families of triangles
论文作者
论文摘要
我们证明，一个家庭的$ \ Mathcal {t} $在$ n $上的$ n $给定的顶点没有彩虹三角形（也就是说，三个边缘，每个边缘从$ \ Mathcal {t} $中的另一个三角形中取。我们还表明，这个结果是尖锐的，并且表征了极端情况。此外，我们讨论了此问题的一个版本，其中三角形不一定是不同的，并表明在这种情况下，相同的界限是渐进的。在发布了本文的原始Arxiv版本后，我们了解到，Győri（2006）（2006）证明了$ \ frac {n^2} {8} $的尖锐上限，并由Frankl，Funredi和Simonyi独立（未发表）独立。对于允许重复的情况，Győri还获得了我们结果的更强版本。
We prove that a family $\mathcal{T}$ of distinct triangles on $n$ given vertices that does not have a rainbow triangle (that is, three edges, each taken from a different triangle in $\mathcal{T}$, that form together a triangle) must be of size at most $\frac{n^2}{8}$. We also show that this result is sharp and characterize the extremal case. In addition, we discuss a version of this problem in which the triangles are not necessarily distinct, and show that in this case, the same bound holds asymptotically. After posting the original arXiv version of this paper, we learned that the sharp upper bound of $\frac{n^2}{8}$ was proved much earlier by Győri (2006) and independently by Frankl, Füredi and Simonyi (unpublished). Győri also obtained a stronger version of our result for the case when repetitions are allowed.
