分裂多维项链与欧几里得空间的可测着色
Splitting multidimensional necklaces and measurable colorings of Euclidean spaces
摘要：
Goldberg和West的项链分裂定理断言，任何k色（连续）项链都可以使用最多k个切割公平地分裂。受Erdos问题的启发，Alon等人证明了实数线存在（t+3）-着色，其中没有项链具有最多使用t切割的公平分裂。我们将这一结果推广到高维空间。更具体地说，我们证明了存在R^{d}的k-染色，使得没有立方体具有大小为t的公平分裂（使用最多t个与每个轴正交的超平面），条件是k>（t+4）^{d}-（t+3）^{d}+（t+2）^{d}-2 ^{d}+d（t+2）+3。我们还考虑了一个离散的变种的多维项链分裂问题的精神的定理的de Longueville和Ekivaljević。一个d维k色立方体的公平分裂需要多少个轴对齐的超平面的问题仍然是开放的。
英文摘要：
A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erdős on strongly nonrepetitive sequences, Alon et al. proved that there is a (t+3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R^{d} such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k>(t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and Živaljević. The question how many axes aligned hyperplanes are needed for a fair splitting of a d-dimensional k-colored cube remains open.
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