Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

Grytczuk, Jarosław; Lubawski, Wojciech

doi:10.48550/arxiv.1209.1809

Cited by 2 publications

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“…For every integers k, t ≥ 1, if k > t + 2, then there exists a measurable k-coloring of R such that no necklace has a fair 2-splitting using at most t cuts. [7] generalized our result to higher dimensional spaces, by proving that the same holds in R d (for q = 2), provided…”

Section: Introductionmentioning

confidence: 61%

“…, k, and for each of them we need at least d q 2 cuts, so in total at least d(k − 1) q 2 cuts. The gap between an upper and a lower bound from Theorem 6.1 motivates the following generalization of the question of Grytczuk and Lubawski [7]. Question 6.2.…”

Section: A Discrete Casementioning

confidence: 99%

“…They expected the lower bound on the number of colors is far from optimal, as even for d = 1 it gives much worse result than the one from [3]. Grytczuk and Lubawski asked [7,Problem 9] if k ≥ t + O(d) colors suffice for the case of two parts (q = 2).…”

Section: Grytczuk and Lubawskimentioning

confidence: 99%

“…We end the paper with a brief look at a discrete variant of the multidimensional necklace splitting problem, as introduced in [7]. It turns out that the equivalence between discrete and continuous case does not hold in dimensions higher than one.…”

Section: Grytczuk and Lubawskimentioning

confidence: 99%

“…Alon's necklace splitting theorem [2] states that every k-colored discrete 1-dimensional necklace has a fair q-splitting using at most k(q − 1) cuts. Grytczuk and Lubawski [7] asked what is the minimum number of axis-aligned hyperplane cuts that are needed to 2-split fairly any k-colored discrete d-dimensional necklace. At first glance, it is not clear why this number is finite.…”

Section: A Discrete Casementioning

confidence: 99%

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Obstacles for splitting multidimensional necklaces

Lasoń

2015

Proc. Amer. Math. Soc.

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The well-known "necklace splitting theorem" of Alon [2] asserts that every k-colored necklace can be fairly split into q parts using at most t cuts, provided k(q − 1) ≤ t. In a joint paper with Alon et al. [3] we studied a kind of opposite question. Namely, for which values of k and t there is a measurable k-coloring of the real line such that no interval has a fair splitting into 2 parts with at most t cuts? We proved that k > t + 2 is a sufficient condition (while k > t is necessary).We generalize this result to Euclidean spaces of arbitrary dimension d, and to arbitrary number of parts q. We prove that if k(q − 1) > t + d + q − 1, then there is a measurable k-coloring of R d such that no axis-aligned cube has a fair q-splitting using at most t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k(q − 1) > t implied by [2]. Moreover for d = 1, q = 2 we get exactly the result of [3].Additionally, we prove that if a stronger inequality k(q −1) > dt+d+q −1 is satisfied, then there is a measurable k-coloring of R d with no axis-aligned cube having a fair q-splitting using at most t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields.2010 Mathematics Subject Classification. 05D99, 54H99, 12E99, 52C45.

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Section: Introductionmentioning

confidence: 61%

Section: A Discrete Casementioning

confidence: 99%

Section: Grytczuk and Lubawskimentioning

confidence: 99%

Section: Grytczuk and Lubawskimentioning

confidence: 99%

Section: A Discrete Casementioning

confidence: 99%

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Obstacles for splitting multidimensional necklaces

Lasoń

2015

Proc. Amer. Math. Soc.

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Measurable patterns, necklaces, and sets indiscernible by measure

Vrećica¹,

Živaljević²

2013

Preprint

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In some recent papers the classical 'splitting necklace theorem' is linked in an interesting way with a geometric 'pattern avoidance problem', see Alon et al. (Proc. Amer. Math. Soc., 2009), Lubawski (arXiv:1209.1809 [math.CO]), and Lasoń (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of R d such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lasoń, we show that for every collection µ 1 , ..., µ 2d−1 of 2d − 1 continuous finite measures on R d , there exist two nontrivial axis-aligned d-dimensional cuboids (rectangular parallelepipeds) C 1 and C 2 such that µ i (C 1 ) = µ i (C 2 ) for each i ∈ {1, ..., 2d − 1}. We also show by examples that the bound 2d − 1 cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.

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Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

Cited by 2 publications

References 0 publications

Obstacles for splitting multidimensional necklaces

Obstacles for splitting multidimensional necklaces

Measurable patterns, necklaces, and sets indiscernible by measure

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