SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 91-104, January 2021.
Haviv [European J. Combin., 81 (2019), pp. 84--97] has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb{R}$. We show that this actually holds for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\mathbb{R}$---an important graph invariant from coding theory---and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.

中文翻译：

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任何字段上图表示的拓扑边界

SIAM Journal on Discrete Mathematics，第 35 卷，第 1 期，第 91-104 页，2021 年 1 月。
Haviv [European J. Combin., 81 (2019), pp. 84--97] 最近证明了图的色数也是它们在 $\mathbb{R}$ 上的正交维数的下限。我们表明这实际上适用于所有已知的拓扑下界和所有场。我们还改进了他在 $\mathbb{R}$ 上为 minrank 参数获得的拓扑边界——编码理论中的一个重要的图不变量——并表明这个边界实际上也适用于所有领域。拟阵上的独立表示的概念被引入并用于将这些结果作为推论的一般定理。还讨论了相关的复杂性结果。