This paper extends the scenario of the Four Color Theorem in the following way. Let $$\fancyscript{H}_{d,k}$$Hd,k be the set of all $$k$$k-uniform hypergraphs that can be (linearly) embedded into $$\mathbb {R}^d$$Rd. We investigate lower and upper bounds on the maximum (weak) chromatic number of hypergraphs in $$\fancyscript{H}_{d,k}$$Hd,k. For example, we can prove that for $$d\ge 3$$d≥3 there are hypergraphs in $$\fancyscript{H}_{2d-3,d}$$H2d-3,d on $$n$$n vertices whose chromatic number is $$\Omega (\log n/\log \log n)$$Ω(logn/loglogn), whereas the chromatic number for $$n$$n-vertex hypergraphs in $$\fancyscript{H}_{d,d}$$Hd,d is bounded by $${\mathcal {O}}(n^{(d-2)/(d-1)})$$O(n(d-2)/(d-1)) for $$d\ge 3$$d≥3.

中文翻译：

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着色 $$d$$ d -可嵌入 $$k$$ k -统一超图


本文通过以下方式扩展了四色定理的场景。令 $$\fancyscript{H}_{d,k}$$Hd,k 为所有可（线性）嵌入到 $$\mathbb {R}^d 中的 $$k$$k 一致超图的集合$$路。我们研究 $$\fancyscript{H}_{d,k}$$Hd,k 中超图最大（弱）色数的下界和上限。例如，我们可以证明对于$$d\ge 3$$d≥3，在$$n$上的$$\fancyscript{H}_{2d-3,d}$$H2d-3,d中存在超图$n 个顶点的色数为 $$\Omega (\log n/\log \log n)$$Ω(logn/loglogn)，而 $$\fancyscript 中 $$n$$n 顶点超图的色数{H}_{d,d}$$Hd,d 的边界为 $${\mathcal {O}}(n^{(d-2)/(d-1)})$$O(n(d -2)/(d-1)) 对于 $$d\ge 3$$d≥3。