We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element i in each color class there exists a triple which does not contain i. We give a linear (in the number of triples) time algorithm to decide if such a coloring exists and find one if it does. We also consider generalizations of this result and an application to a matching problem, which motivated this study. Finally, we show how these results translate to results about colorings of hypergraphs in which the degree of every vertex is k less than the number of hyperedges.

中文翻译：

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两种颜色的三元组，这样，在每个颜色类别中，每个元素至少都会丢失一次

我们给出了可以用两种颜色上色的元素（例如，正整数）三元组的有限集合的特征，这样对于每种颜色类别中的每个元素i，都存在一个不包含i的三元组。我们给出一个线性（以三元组数为单位）的时间算法来确定是否存在这种着色，并找到是否存在。我们还考虑了该结果的概括以及对匹配问题的应用，这激发了这项研究。最后，我们显示这些结果如何转换为有关超图着色的结果，其中每个顶点的度数比超边数少k。