It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3-coloring of the edges? Gyarfas proved that $(2n+3)/3$ is an absolute lower bound and that this lower bound is best possible for infinitely many $n$. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually $(1-o(1))n$. We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest \emph{3-partite hole} (that is, sets $X_1, X_2, X_3$ with $|X_1|=|X_2|=|X_3|$ such that no edge intersects all of $X_1, X_2, X_3$) in the Steiner triple system (Gyarfas previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the coloring has a particular structure. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.

中文翻译：

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三边色 Steiner 三重系统中的大单色分量

已知在完全 $r$-uniform 超图的边的任何 $r$-coloring 中，都存在一个跨越单色分量。给定 $n$ 个顶点上的 Steiner 三重系统，在边缘的任意 3 种着色中可以保证的最大单色分量是什么？Gyarfas 证明了 $(2n+3)/3$ 是一个绝对下限，并且这个下限对于无限多的 $n$ 是最好的。另一方面，我们证明了几乎所有 Steiner 三元系统的下限实际上是 $(1-o(1))n$。我们得到这个结果作为一个更一般的定理的结果，该定理表明下界取决于最大的 \emph{3-partite hole}（即，将 $X_1, X_2, X_3$ 设置为 $|X_1| =|X_2|=|X_3|$ 使得没有边与所有 $X_1、X_2、X_3$) 在 Steiner 三重系统中（Gyarfas 之前观察到上限取决于此参数）。此外，我们表明，除非着色具有特定结构，否则该下限是紧密的。我们还建议在 Steiner 三重系统的设置中解决各种其他 Ramsey 问题。