Geometric motivations warranted the study of hypergraphs on ordered vertices that have no pair of hyperedges that induce an alternation of some given length. Such hypergraphs are called ABA-free, ABAB-free and so on. Since then various coloring and other combinatorial results were proved about these families of hypergraphs. We prove a characterization in terms of their incidence matrices which avoids using the ordering of the vertices. Using this characterization, we prove new results about the dual hypergraphs of ABAB-free hypergraphs. In particular, we show that dual-ABAB-free hypergraphs are not always proper $2$-colorable even if we restrict ourselves to hyperedges that are larger than some parameter $m$.

翻译：几何动机促使我们研究有序顶点上的超图，这些超图不存在能诱导出特定长度交替的超边对。此类超图被称为ABA自由、ABAB自由等。此后，关于这些超图族已有各种着色及其他组合结果被证明。我们提出了一种基于关联矩阵的刻画方法，该方法避免使用顶点序。利用这一刻画，我们证明了对偶-ABAB自由超图的新结果。特别地，我们证明即使将超边限制为大于某参数$m$，对偶-ABAB自由超图也并非总是可正常$2$着色的。